Answer to Question #198871 in Calculus for Snakho

Question

y(x) =(sinx) linx

Answer 1

$$y=(\sin x)^{\ln x}$$

Then

$$\ln y=\ln{(\sin x)^{\ln x}}$$

$$\ln y=\ln x\cdot\ln{(\sin x)}$$

Differentiate both sides with respect to $x$

$$\dfrac{d}{dx}(\ln y)=\dfrac{d}{dx}(\ln x\cdot\ln{(\sin x)})$$

Use the Chain Rule

$$\dfrac{1}{y}\cdot\dfrac{dy}{dx}=\dfrac{\ln (\sin x)}{x}+\dfrac{\ln x}{\sin x}\cdot \cos x$$

$$\dfrac{dy}{dx}=y(\dfrac{\ln (\sin x)}{x}+\dfrac{\ln x}{\tan x})$$

$$\dfrac{dy}{dx}=(\sin x)^{\ln x}\bigg(\dfrac{\ln (\sin x)}{x}+\dfrac{\ln x}{\tan x}\bigg)$$