Question

Find the derivative of ln(cosx²).​

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} = -2x \tan (x^2)$

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]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle y = \ln (\cos x^2)$

Step 2: Differentiate

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

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

Unit: Differentiation