Question

Derivative of $y=(sinx{)}^2$ with respect to x will be

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

All the formulae mentioned for the derivative of standard functions hold true only when the variable is x itself, not for any other function of x. To make it more clear let’s take lnx as the function. Its derivative with respect to x as we know will be
$\frac{1}{x}$, but the derivative of lnsinx with respect to x will not be $\frac{1}{sinx}$. Rather it will be $\frac{1}{sinx}$ multiplied by $\frac{d\left(sinx\right)}{dx}$. Yes the derivative of lnsinx with respect to sinx will be $\frac{1}{sinx}$ but not with respect to x
In this question the function is of the form of $x^n$. The only difference is that it has sinx at the place of x. So we will use chain rule.
$.\frac{dy}{dx}=\frac{dy}{d\left(sinx\right)}\times \frac{d\left(sinx\right)}{dx}$
$\frac{dy}{d\left(sinx\right)}=2sinx$
$\frac{d\left(sinx\right)}{dx}=cosx$
So $\frac{dy}{dx}=2sinxcosx$

Also you can write the function as sinx.sinx and try to solve it using product rule.