Question

Find the derivative of the trigonometric function

Answer 1

E.
`\displaystyle f'(s) = s^5 \sec^2 s + 5s^4 \tan x`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle f(s) = s^5 \tan s`

Step 2: Differentiate

1. Derivative Rule [Product Rule]:
   `\displaystyle f'(s) = (s^5)' \tan s + s^5(\tan s)'`
2. Basic Power Rule/Trigonometric Differentiation:
   `\displaystyle f'(s) = 5s^4 \tan s + s^5 \sec^2 x`
3. Rewrite:
   `\displaystyle f'(s) = s^5 \sec^2 s + 5s^4 \tan x`

∴ Our answer is E.

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation