Question

Find the derivative of ln(cosx²).​

Answer 1

`\displaystyle (dy)/(dx) = -2x \tan (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 = \ln (\cos x^2)`

Step 2: Differentiate

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

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

Unit: Differentiation