Question

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

Answer 1

In the given problem, we are required to differentiate $\left( {{{\sin }^2}x} \right)$ with respect to x. Since, $\left( {{{\sin }^2}x} \right)$ is a composite function, so we will have to apply chain rule of differentiation in the process of differentiating $\left( {{{\sin }^2}x} \right)$ . So, differentiation of $\left( {{{\sin }^2}x} \right)$ with respect to x will be done layer by layer using the chain rule of differentiation. Also the derivative of $\sin (x)$with respect to $x$ must be remembered.

Complete step by step answer:
So, Derivative of $\left( {{{\sin }^2}x} \right)$ with respect to $x$can be calculated as $\dfrac{d}{{dx}}\left( {{{\sin }^2}x} \right)$ .
Now, $\dfrac{d}{{dx}}\left( {{{\sin }^2}x} \right)$
Taking the power outside the bracket in order to apply chain rule of differentiation.
$\dfrac{d}{{dx}}\left[ {{{\left( {\sin x} \right)}^2}} \right]$
Now, Let us assume $u = \sin (x)$. So substituting $\sin (x)$ as $u$, we get,
$\dfrac{d}{{dx}}{\left[ u \right]^2} = 2u\dfrac{{du}}{{dx}}$

Now, putting back $u$ as $\sin (x)$, we get,
$2\sin x\dfrac{{d\left( {\sin x} \right)}}{{dx}}$ because $\dfrac{{du}}{{dx}} = \dfrac{{d(\sin x)}}{{dx}}$
Now, we know that the derivative of $\sin x$ with respect to $x$ is $\cos x$. So, $\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x$.
So, Substituting the equivalent expression of $\dfrac{d}{{dx}}\left( {\sin x} \right)$, we get,
$2\sin x\left( {\cos x} \right)$
Now, we know the double angle formula for sine function as $\sin 2x = 2\sin x\cos x$. So, we get $\sin 2x$.

So, the derivative of $\left( {{{\sin }^2}x} \right)$ with respect to $x$ is $\sin 2x$.

Note: The derivatives of basic trigonometric functions must be learned by the heart in order to find derivatives of complex composite functions using the chain rule of differentiation. The chain rule of differentiation involves differentiating a composite by introducing new unknowns to ease the process and examine the behavior of function layer by layer. The answer to the problem can also be $2\sin x\left( {\cos x} \right)$, but it is better to use the double angle formula of the sine function and give a precise final answer.