- The function $ y = x^{\sin x} $ has a variable base and a variable exponent, so we use logarithmic differentiation.
- Take the natural logarithm of both sides: $ \ln y = \ln(x^{\sin x}) = \sin x \cdot \ln x $.
- Differentiate both sides implicitly with respect to $ x $:
- Left side: $ \frac{d}{dx}[\ln y] = \frac{1}{y} \cdot \frac{dy}{dx} $.
- Right side: Use the product rule on $ \sin x \cdot \ln x $:
$ \frac{d}{dx}[\sin x \cdot \ln x] = \cos x \cdot \ln x + \sin x \cdot \frac{1}{x} $.
- Combine the results: $ \frac{1}{y} \cdot \frac{dy}{dx} = \cos x \cdot \ln x + \frac{\sin x}{x} $.
- Solve for $ \frac{dy}{dx} $: $ \frac{dy}{dx} = y \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) $.
- Substitute back $ y = x^{\sin x} $: $ \frac{dy}{dx} = x^{\sin x} \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) $.
Parent Tip: Review the logic above to help your child master the concept of calculus review worksheet.
