Question

Differentiate ${\sin ^2}\left( {{x^2}} \right)$ with respect to ${x^2}$

Answer 1

:We will make use of the standard formula which says As we know that $2{\text{ }}sinx{\text{ }}.{\text{ }}cosx{\text{ }} = {\text{ }}sin{\text{ }}2x$. Further we will differentiate the value ${\sin ^2}({x^2})$.

Complete step by step solution:
Consider the given differentiation
$y = {\sin ^2}\left( {{x^2}} \right)$
Differentiate with respect to ${x^2}$
$\dfrac{{dy}}{{d{x^2}}} = \dfrac{{d{{\sin }^2}{x^2}}}{{d{x^2}}}$
$\dfrac{{dy}}{{d{x^2}}} = 2\sin {x^2}\cos {x^2}$
As we know that $2{\text{ }}sin{\text{ }}xcos{\text{ }}x = sin{\text{ }}2x{\text{ }}so,$
$2\sin {x^2}\cos {x^2} = \sin 2{x^2}$
$\dfrac{{dy}}{{d{x^2}}} = \sin \left( {2{x^2}} \right)$
Hence, this is the answer

Note: We can also solve these problem by taking $f = {\sin ^2}{x^2}\,\,g = {x^2}$ to get
$\dfrac{{df}}{{dg}} = \dfrac{{df}}{{db}} \times \dfrac{{db}}{{dg}} = \dfrac{{df/db}}{{db/dg}}$