# How do you find the derivative of y=(sinx)^(x^3)?

Jun 12, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{2} {\left(\sin x\right)}^{{x}^{3}} \left({x}^{2} \ln \sin x + x \cot x\right)$

#### Explanation:

$y = {\left(\sin x\right)}^{{x}^{3}}$

take natural logs

$\ln y = \ln {\left(\sin x\right)}^{{x}^{3}}$

$\implies \ln y = {x}^{3} \ln \sin x$

$\frac{d}{\mathrm{dx}} \left(\ln y\right) = \frac{d}{\mathrm{dx}} \left({x}^{3} \ln \sin x\right)$

differentiate wrt$x$

RHS using the product rule

$\frac{1}{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 3 {x}^{2} \ln \sin x + {x}^{3} \cos \frac{x}{\sin} x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = y \left[{x}^{2} \left(3 \ln \sin x + x \cot x\right)\right]$

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{2} {\left(\sin x\right)}^{{x}^{3}} \left({x}^{2} \ln \sin x + x \cot x\right)$