Question

Derivative of y=ln(sinx^3)

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} = 3x^2 \cot x^3$

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

$\displaystyle y = \ln (\sin x^3)$

Step 2: Differentiate

1. Logarithmic Differentiation [Derivative Rule - Chain Rule]:                       $\displaystyle y' = \frac{1}{\sin x^3} \cdot \frac{d}{dx}[\sin x^3]$
2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:                   $\displaystyle y' = \frac{\cos x^3}{\sin x^3} \cdot \frac{d}{dx}[x^3]$
3. Simplify:                                                                                                        $\displaystyle y' = \cot x^3 \cdot \frac{d}{dx}[x^3]$
4. Basic Power Rule:                                                                                         $\displaystyle y' = 3x^2 \cot x^3$

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

Unit: Differentiation