Answer to Question #214141 in Calculus for sbuda

Question

Differentiate the following with regard to x :

y=x^4sinx

Answer 1

Given the equation

$y=x^{4sinx}$

The derivative of the function is obtained thus:

$\begin{aligned} &\begin{gathered} \quad \frac{\mathrm{d}}{\mathrm{d} x}\left[x^{4 \sin (x)}\right] \\ =x^{4 \sin (x)} \cdot \frac{\mathrm{d}}{\mathrm{d} x}[\ln (x) \cdot 4 \sin (x)] \\ =x^{4 \sin (x)} \cdot 4 \cdot \frac{\mathrm{d}}{\mathrm{d} x}[\ln (x) \sin (x)] \\ =4 x^{4 \sin (x)}\left(\frac{\mathrm{d}}{\mathrm{d} x}[\ln (x)] \cdot \sin (x)+\ln (x) \cdot \frac{\mathrm{d}}{\mathrm{d} x}[\sin (x)]\right) \\ =4 x^{4 \sin (x)}\left(\frac{1}{x} \sin (x)+\ln (x) \cos (x)\right) \\ =4 x^{4 \sin (x)}\left(\frac{\sin (x)}{x}+\cos (x) \ln (x)\right) \end{gathered}\\ &\text { We can rewrite the above result as: }\\ &=x^{4 \sin (x)}\left(\frac{4 \sin (x)}{x}+4 \cos (x) \ln (x)\right) \end{aligned}$