Question

Find the derivative of the following function (n is an integer).
${\sin }^{n}x$

Answer 1

 Hint: - Apply chain rule.
${\sin }^{a}xdx=a{\sin }^{a-1}x.\left(\frac{d}{dx}\sin x\right)$
As we know the differentiation of $\sin x$ is $\cos x$ and using the formula which is stated above after the application of chain rule for the differentiation of ${\sin }^{n}x$is

$$\begin{gathered} \Rightarrow {\sin }^{n}xdx=n{\sin }^{n-1}x\left(\frac{d}{dx}\sin x\right) \\ =n\text{si}{\text{n}}^{n-1}x\left(\cos x\right) \\ =n\cos x{\sin }^{n-1}x \end{gathered}$$
So, this is the required differentiation of ${\sin }^{n}x$
Note: - Such problems demand the usage of chain rule and for solving them always remember the basic differentiation formulas. Remember that ${\sin }^{n}x$ is not the same as $\sin nx$ to avoid mistakes.