Question

Derivative of y=ln(sinx^3)

Answer 1

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

Explanation:

Step 1: Define

Identify

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

Step 2: Differentiate

1. Logarithmic Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = (1)/(\sin x^3) \cdot (d)/(dx)[\sin x^3]`
2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = (\cos x^3)/(\sin x^3) \cdot (d)/(dx)[x^3]`
3. Simplify:
   `\displaystyle y' = \cot x^3 \cdot (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