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

$\Rightarrow {\sin ^n}xdx = n{\sin ^{n - 1}}x\left( {\frac{d}{{dx}}\sin x} \right) \\\ {\text{ }} = n{\text{si}}{{\text{n}}^{n - 1}}x\left( {\cos x} \right) \\\ {\text{ }} = n\cos x{\sin ^{n - 1}}x \\\$
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 ${\sin n}x$ to avoid mistakes.