Question

Find the derivative of cos^4(5x^2)

Answer 1

`\displaystyle (dy)/(dx) = -40x \sin (5x^2) \cos^3 (5x^2)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = \cos^4 (5x^2)`

Step 2: Differentiate

1. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle y' = 4 \cos^3 (5x^2)[\cos (5x^2)]'`
2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = 4 \cos^3 (5x^2) \sin (5x^2) (5x^2)'`
3. Basic Power Rule [Derivative Property - Multiplied Constant]:
   `\displaystyle y' = -40x \sin (5x^2) \cos^3 (5x^2)`

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

Unit: Differentiation