Basic Trigonometric Derivative — Dynamic Practice
Basic Trigonometric Derivative — Dynamic Practice. Practice questions to deepen understanding of basic trigonometric derivatives. Online math practice with full solutions and step-by-step explanations.
Dynamic practice in basic trigonometric derivatives — (sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = sec²x. New questions every attempt.
Question 1
2.38 pts
Find the derivative of the function:
\(f(x) = cos(x-4)\)
\(f(x) = cos(x-4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(x-4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -\sin(x-4)\)
The function: \(f(x) = cos(x-4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x-4\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = -\sin(x-4)\)
Question 2
2.38 pts
Find the derivative of the function:
\(f(x) = sin(x)\)
\(f(x) = sin(x)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(x)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \cos(x)\)
The function: \(f(x) = sin(x)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \cos(x)\)
Question 3
2.38 pts
Find the derivative of the function:
\(f(x) = sin(x+1)\)
\(f(x) = sin(x+1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(x+1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \cos(x+1)\)
The function: \(f(x) = sin(x+1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+1\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \cos(x+1)\)
Question 4
2.38 pts
Find the derivative of the function:
\(f(x) = sin(x-5)\)
\(f(x) = sin(x-5)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(x-5)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \cos(x-5)\)
The function: \(f(x) = sin(x-5)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x-5\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \cos(x-5)\)
Question 5
2.38 pts
Find the derivative of the function:
\(f(x) = cos(3x-1)\)
\(f(x) = cos(3x-1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(3x-1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -3\sin(3x-1)\)
The function: \(f(x) = cos(3x-1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x-1\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = -3\sin(3x-1)\)
Question 6
2.38 pts
Find the derivative of the function:
\(f(x) = cos(x-2)\)
\(f(x) = cos(x-2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(x-2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -\sin(x-2)\)
The function: \(f(x) = cos(x-2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x-2\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = -\sin(x-2)\)
Question 7
2.38 pts
Find the derivative of the function:
\(f(x) = cos(x+1)\)
\(f(x) = cos(x+1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(x+1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -\sin(x+1)\)
The function: \(f(x) = cos(x+1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+1\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = -\sin(x+1)\)
Question 8
2.38 pts
Find the derivative of the function:
\(f(x) = tan(x+4)\)
\(f(x) = tan(x+4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(x+4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{1}{\cos^2(x+4)}\)
The function: \(f(x) = tan(x+4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+4\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \frac{1}{\cos^2(x+4)}\)
Question 9
2.38 pts
Find the derivative of the function:
\(f(x) = tan(3x-4)\)
\(f(x) = tan(3x-4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(3x-4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{3}{\cos^2(3x-4)}\)
The function: \(f(x) = tan(3x-4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x-4\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = \frac{3}{\cos^2(3x-4)}\)
Question 10
2.38 pts
Find the derivative of the function:
\(f(x) = cos(x+4)\)
\(f(x) = cos(x+4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(x+4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -\sin(x+4)\)
The function: \(f(x) = cos(x+4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+4\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = -\sin(x+4)\)
Question 11
2.38 pts
Find the derivative of the function:
\(f(x) = sin(3x-1)\)
\(f(x) = sin(3x-1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(3x-1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = 3\cos(3x-1)\)
The function: \(f(x) = sin(3x-1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x-1\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = 3\cos(3x-1)\)
Question 12
2.38 pts
Find the derivative of the function:
\(f(x) = cos(2x+3)\)
\(f(x) = cos(2x+3)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(2x+3)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -2\sin(2x+3)\)
The function: \(f(x) = cos(2x+3)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(2x+3\) |
| 🟠 Inner derivative: | \(2\) |
Answer: \(f'(x) = -2\sin(2x+3)\)
Question 13
2.38 pts
Find the derivative of the function:
\(f(x) = tan(x-4)\)
\(f(x) = tan(x-4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(x-4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{1}{\cos^2(x-4)}\)
The function: \(f(x) = tan(x-4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x-4\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \frac{1}{\cos^2(x-4)}\)
Question 14
2.38 pts
Find the derivative of the function:
\(f(x) = cos(3x-5)\)
\(f(x) = cos(3x-5)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(3x-5)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -3\sin(3x-5)\)
The function: \(f(x) = cos(3x-5)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x-5\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = -3\sin(3x-5)\)
Question 15
2.38 pts
Find the derivative of the function:
\(f(x) = sin(4x-4)\)
\(f(x) = sin(4x-4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(4x-4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = 4\cos(4x-4)\)
The function: \(f(x) = sin(4x-4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x-4\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = 4\cos(4x-4)\)
Question 16
2.38 pts
Find the derivative of the function:
\(f(x) = tan(2x-2)\)
\(f(x) = tan(2x-2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(2x-2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{2}{\cos^2(2x-2)}\)
The function: \(f(x) = tan(2x-2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(2x-2\) |
| 🟠 Inner derivative: | \(2\) |
Answer: \(f'(x) = \frac{2}{\cos^2(2x-2)}\)
Question 17
2.38 pts
Find the derivative of the function:
\(f(x) = sin(4x+2)\)
\(f(x) = sin(4x+2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(4x+2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = 4\cos(4x+2)\)
The function: \(f(x) = sin(4x+2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x+2\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = 4\cos(4x+2)\)
Question 18
2.38 pts
Find the derivative of the function:
\(f(x) = tan(3x+4)\)
\(f(x) = tan(3x+4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(3x+4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{3}{\cos^2(3x+4)}\)
The function: \(f(x) = tan(3x+4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x+4\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = \frac{3}{\cos^2(3x+4)}\)
Question 19
2.38 pts
Find the derivative of the function:
\(f(x) = tan(3x+3)\)
\(f(x) = tan(3x+3)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(3x+3)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{3}{\cos^2(3x+3)}\)
The function: \(f(x) = tan(3x+3)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x+3\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = \frac{3}{\cos^2(3x+3)}\)
Question 20
2.38 pts
Find the derivative of the function:
\(f(x) = cos(4x+5)\)
\(f(x) = cos(4x+5)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(4x+5)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -4\sin(4x+5)\)
The function: \(f(x) = cos(4x+5)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x+5\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = -4\sin(4x+5)\)
Question 21
2.38 pts
Find the derivative of the function:
\(f(x) = cos(x+2)\)
\(f(x) = cos(x+2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(x+2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -\sin(x+2)\)
The function: \(f(x) = cos(x+2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+2\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = -\sin(x+2)\)
Question 22
2.38 pts
Find the derivative of the function:
\(f(x) = tan(4x+3)\)
\(f(x) = tan(4x+3)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(4x+3)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{4}{\cos^2(4x+3)}\)
The function: \(f(x) = tan(4x+3)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x+3\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = \frac{4}{\cos^2(4x+3)}\)
Question 23
2.38 pts
Find the derivative of the function:
\(f(x) = tan(4x-2)\)
\(f(x) = tan(4x-2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(4x-2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{4}{\cos^2(4x-2)}\)
The function: \(f(x) = tan(4x-2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x-2\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = \frac{4}{\cos^2(4x-2)}\)
Question 24
2.38 pts
Find the derivative of the function:
\(f(x) = tan(4x-4)\)
\(f(x) = tan(4x-4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(4x-4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{4}{\cos^2(4x-4)}\)
The function: \(f(x) = tan(4x-4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x-4\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = \frac{4}{\cos^2(4x-4)}\)
Question 25
2.38 pts
Find the derivative of the function:
\(f(x) = tan(x+2)\)
\(f(x) = tan(x+2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(x+2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{1}{\cos^2(x+2)}\)
The function: \(f(x) = tan(x+2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+2\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \frac{1}{\cos^2(x+2)}\)
Question 26
2.38 pts
Find the derivative of the function:
\(f(x) = cos(3x+1)\)
\(f(x) = cos(3x+1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(3x+1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -3\sin(3x+1)\)
The function: \(f(x) = cos(3x+1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x+1\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = -3\sin(3x+1)\)
Question 27
2.38 pts
Find the derivative of the function:
\(f(x) = cos(4x+2)\)
\(f(x) = cos(4x+2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(4x+2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -4\sin(4x+2)\)
The function: \(f(x) = cos(4x+2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x+2\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = -4\sin(4x+2)\)
Question 28
2.38 pts
Find the derivative of the function:
\(f(x) = cos(4x-4)\)
\(f(x) = cos(4x-4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(4x-4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -4\sin(4x-4)\)
The function: \(f(x) = cos(4x-4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x-4\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = -4\sin(4x-4)\)
Question 29
2.38 pts
Find the derivative of the function:
\(f(x) = sin(5x+1)\)
\(f(x) = sin(5x+1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(5x+1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = 5\cos(5x+1)\)
The function: \(f(x) = sin(5x+1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x+1\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = 5\cos(5x+1)\)
Question 30
2.38 pts
Find the derivative of the function:
\(f(x) = cos(5x-2)\)
\(f(x) = cos(5x-2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(5x-2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -5\sin(5x-2)\)
The function: \(f(x) = cos(5x-2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x-2\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = -5\sin(5x-2)\)
Question 31
2.38 pts
Find the derivative of the function:
\(f(x) = tan(4x+1)\)
\(f(x) = tan(4x+1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(4x+1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{4}{\cos^2(4x+1)}\)
The function: \(f(x) = tan(4x+1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x+1\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = \frac{4}{\cos^2(4x+1)}\)
Question 32
2.38 pts
Find the derivative of the function:
\(f(x) = sin(x-1)\)
\(f(x) = sin(x-1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(x-1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \cos(x-1)\)
The function: \(f(x) = sin(x-1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x-1\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \cos(x-1)\)
Question 33
2.38 pts
Find the derivative of the function:
\(f(x) = tan(x+5)\)
\(f(x) = tan(x+5)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(x+5)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{1}{\cos^2(x+5)}\)
The function: \(f(x) = tan(x+5)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(x+5\) |
| 🟠 Inner derivative: | \(1\) |
Answer: \(f'(x) = \frac{1}{\cos^2(x+5)}\)
Question 34
2.38 pts
Find the derivative of the function:
\(f(x) = cos(5x-3)\)
\(f(x) = cos(5x-3)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(5x-3)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -5\sin(5x-3)\)
The function: \(f(x) = cos(5x-3)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x-3\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = -5\sin(5x-3)\)
Question 35
2.38 pts
Find the derivative of the function:
\(f(x) = sin(3x+4)\)
\(f(x) = sin(3x+4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = sin(3x+4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = 3\cos(3x+4)\)
The function: \(f(x) = sin(3x+4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(3x+4\) |
| 🟠 Inner derivative: | \(3\) |
Answer: \(f'(x) = 3\cos(3x+4)\)
Question 36
2.38 pts
Find the derivative of the function:
\(f(x) = tan(2x-5)\)
\(f(x) = tan(2x-5)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(2x-5)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{2}{\cos^2(2x-5)}\)
The function: \(f(x) = tan(2x-5)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(2x-5\) |
| 🟠 Inner derivative: | \(2\) |
Answer: \(f'(x) = \frac{2}{\cos^2(2x-5)}\)
Question 37
2.38 pts
Find the derivative of the function:
\(f(x) = tan(4x-1)\)
\(f(x) = tan(4x-1)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(4x-1)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{4}{\cos^2(4x-1)}\)
The function: \(f(x) = tan(4x-1)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(4x-1\) |
| 🟠 Inner derivative: | \(4\) |
Answer: \(f'(x) = \frac{4}{\cos^2(4x-1)}\)
Question 38
2.38 pts
Find the derivative of the function:
\(f(x) = tan(2x+2)\)
\(f(x) = tan(2x+2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(2x+2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{2}{\cos^2(2x+2)}\)
The function: \(f(x) = tan(2x+2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(2x+2\) |
| 🟠 Inner derivative: | \(2\) |
Answer: \(f'(x) = \frac{2}{\cos^2(2x+2)}\)
Question 39
2.38 pts
Find the derivative of the function:
\(f(x) = cos(5x-5)\)
\(f(x) = cos(5x-5)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(5x-5)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -5\sin(5x-5)\)
The function: \(f(x) = cos(5x-5)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x-5\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = -5\sin(5x-5)\)
Question 40
2.38 pts
Find the derivative of the function:
\(f(x) = cos(5x+4)\)
\(f(x) = cos(5x+4)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(5x+4)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -5\sin(5x+4)\)
The function: \(f(x) = cos(5x+4)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x+4\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = -5\sin(5x+4)\)
Question 41
2.38 pts
Find the derivative of the function:
\(f(x) = tan(5x-2)\)
\(f(x) = tan(5x-2)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = tan(5x-2)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = \frac{5}{\cos^2(5x-2)}\)
The function: \(f(x) = tan(5x-2)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x-2\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = \frac{5}{\cos^2(5x-2)}\)
Question 42
2.38 pts
Find the derivative of the function:
\(f(x) = cos(5x)\)
\(f(x) = cos(5x)\)
Explanation:
Solution – Trigonometric derivatives:
The function: \(f(x) = cos(5x)\)
📊 Trigonometric derivatives:
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
Answer: \(f'(x) = -5\sin(5x)\)
The function: \(f(x) = cos(5x)\)
📊 Trigonometric derivatives:
| \((\sin x)' = \cos x\) | \((\cos x)' = -\sin x\) |
| \((\tan x)' = \frac{1}{\cos^2 x}\) | \((\cot x)' = \frac{-1}{\sin^2 x}\) |
🧅 Onion method:
There is a composition here! Differentiate the outer and multiply by the inner derivative.
| 🔵 Outer layer: | The trigonometric function |
| 🟢 Inner layer: | \(5x\) |
| 🟠 Inner derivative: | \(5\) |
Answer: \(f'(x) = -5\sin(5x)\)