Question

What is \[\frac{dy}{dx}\] for \[y=x^{\sin x}\], \[x>0\]?

Options

• \[x^{\sin x}\left[\frac{\cos x}{x}+\sin x\log x\right]\]

• \[x^{\cos x}\left[\frac{\sin x}{x}+\cos x\log x\right]\]

• \[x^{\sin x}\left[\frac{\sin x}{x}+\cos x\log x\right]\]

• \[x^{\sin x}\left[\sin x+x\cos x\log x\right]\]

Answer 1

Differentiate \[\sin x\log x\] by the product rule. This gives \[\frac{\sin x}{x}+\cos x\log x\], which is then multiplied by \[y=x^{\sin x}\].