Question

What’s the derivative of sin^2(cos(x^2))

Answer 1

`\displaystyle (dy)/(dx) = -4x \sin x^2 \sin (\cos x^2) \cos (\cos x^2)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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 = \sin^2 (\cos x^2)`

Step 2: Differentiate

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

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

Unit: Differentiation