Question

If $y=\cos x^3$, then find $\frac{dy}{dx}$.

Answer 1

This type of questions are done by substitution. Put $x^3=u$, and differentiate $u$ w.r.t $x$. So
$3x^2=\frac{du}{dx}.....\left(i\right)$
Also, $y=\cos u$.
Now, differentiating $y$ w.r.t. $u$, we get
$\frac{dy}{du}=-\sin u=-\sin x^3....\left(ii\right)$
Using Eqs. (i) and (ii), we get $\frac{dy}{dx}\times \frac{du}{dx}=\frac{dy}{dx}=-3x^2\sin x^3$.