Question

Differentiate \( y = \sqrt{\log \sin \left( \frac{x^3}{3} - 1 \right)} \) with respect to \( x \).

Hint:

Quick Tip: When differentiating composite functions like logarithmic and trigonometric functions, use the chain rule. Remember to simplify the expression step-by-step to avoid errors.

Answer 1

Given the function: \[ y = \sqrt{\log \sin \left( \frac{x^3}{3} - 1 \right)} \] Let \( u = \log \sin \left( \frac{x^3}{3} - 1 \right) \). The equation simplifies to: \[ y = \sqrt{u} \] Differentiating both sides with respect to \( x \): \[ \frac{dy}{dx} = \frac{1}{2\sqrt{u}} \frac{du}{dx} \] Differentiate \( u \) with respect to \( x \): \[ u = \log \sin \left( \frac{x^3}{3} - 1 \right) \] \[ \frac{du}{dx} = \frac{1}{\sin \left( \frac{x^3}{3} - 1 \right)} \cdot \cos \left( \frac{x^3}{3} - 1 \right) \cdot \frac{d}{dx} \left( \frac{x^3}{3} - 1 \right) \] \[ \frac{d}{dx} \left( \frac{x^3}{3} - 1 \right) = x^2 \] Therefore, \[ \frac{du}{dx} = \frac{x^2 \cos \left( \frac{x^3}{3} - 1 \right)}{\sin \left( \frac{x^3}{3} - 1 \right)} \] Substituting this back into the expression for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{1}{2 \sqrt{\log \sin \left( \frac{x^3}{3} - 1 \right)}} \cdot \frac{x^2 \cos \left( \frac{x^3}{3} - 1 \right)}{\sin \left( \frac{x^3}{3} - 1 \right)} \] This yields the derivative of \( y \) with respect to \( x \).