Question

Find the derivative of the function given by $f\left(x\right)=\sin \left(x^2\right)$

Answer 1

take ,$g\left(x\right)=\sin x$ , $h\left(x\right)=x^2$

then , $f\left(x\right)=g\left(h\right(x\left)\right)$

since $g\left(x\right)$ and $h\left(x\right)$ are differentiable functions, $f\left(x\right)=g\left(h\right(x\left)\right)$ is also differentiable.

By chain rule , we get

$\frac{df\left(x\right)}{dx}=\frac{dg\left(h\right(x\left)\right)}{dx}=g^9\left(h\right(x\left)\right).h^9\left(x\right)$

$=\left(\cos x^2\right)\times \left(2x\right)$

$=2x\cos x^2$

i.e, $f^9\left(x\right)=2x\cos \left(x^2\right)$