Question

# Differentiate.${\sin ^2}\left( {\ln \,x} \right)$.

Hint: We will suppose the given value$\left( {y = {{\sin }^2}\left( {\ln \,x} \right)} \right)$. Differentiate the given value with respect to x.
$\dfrac{d}{{dx}}\log x = \dfrac{1}{x}$

Complete step by step solution:-
Let $y = {\sin ^2}\left( {\ln x} \right)$
Differentiate both side with respect to x, we will get
$\dfrac{d}{{dx}}y = \dfrac{d}{{dx}}{\sin ^2}\left( {\log x} \right) \\ \dfrac{d}{{dx}}y = 2\sin \left( {\log x} \right)\dfrac{d}{{dx}}\sin \left( {\log x} \right) \\ \dfrac{{dy}}{{dx}} = 2\sin \left( {\log x} \right)\cos \left( {\log x} \right)\dfrac{d}{{dx}}\left( {\log x} \right) \\ \dfrac{{dy}}{{dx}} = 2\sin x\left( {\log x} \right)\cos x\left( {\log x} \right)\dfrac{1}{x}\dfrac{d}{{dx}}x \\ \dfrac{{dy}}{{dx}} = 2\sin x\left( {\log x} \right)\cos x\left( {\log x} \right)\dfrac{1}{x} \times 1 \\ \dfrac{{dy}}{{dx}} = 2\sin x\left( {\log x} \right)\cos x\left( {\log x} \right)\dfrac{1}{x} \\ \dfrac{{dy}}{{dx}} = \dfrac{{2\sin x\left( {\log x} \right)\cos x\left( {\log x} \right)}}{x} \\$
Additional information: Differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable. Some differentiation rule are:
(i) The constant rule: for any fixed real number $c$.$\dfrac{d}{{dx}}\left\{ {c.f(x)} \right\} = c.\dfrac{d}{{dx}}\left\{ {f(x)} \right\}$
(ii) The power rule: $\dfrac{d}{{dx}}\left\{ {{x^n}} \right\} = n{x^{n - 1}}$

Note: Students should follow product rule $\left[ {f\left( x \right)g\left( x \right)} \right]$ when we differentiate this value with respect to x then
$\dfrac{d}{{dx}}\left[ {f\left( x \right)g\left( x \right)} \right] \\ = g\left( n \right)\dfrac{d}{{dx}}\left[ {f\left( x \right)} \right] + f\left( x \right)\dfrac{d}{{dx}}\left[ {g\left( x \right)} \right] \\$
DO not differentiate directly.