What is the derivative of (sin x)^(3x)?

Dec 6, 2016

$3 {\left(\sin x\right)}^{3 x} \left(\ln \sin x + x \cot x\right)$

Explanation:

$y = {\left(\sin x\right)}^{3 x}$
taking ln both sides
$\ln y = \ln {\left(\sin x\right)}^{3 x}$
$\ln y = \left(3 x\right) \ln \left(\sin x\right)$
Differentiate both sides
$\left(\frac{1}{y}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 3 \ln \sin x + 3 x \left(\frac{1}{\sin} x\right) \cos x$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(3 \ln \sin x + 3 x \cot x\right) y$
putting value of y
$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 \left(\ln \sin x + x \cot x\right) {\left(\sin x\right)}^{3 x}$
it can be written as
$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {\left(\sin x\right)}^{3 x} \left(\ln \sin x + x \cot x\right)$