Question

How do you find the derivative of ${\sin }^2\left(2x\right)$ .

Answer 1

Hint: We will use different differentiation rules and formulas to solve the given question. As the given function is a composite function so we will use the chain rule to find the derivative. First we will differentiate the function by using power rule ${\displaystyle \frac{d}{dx}}\left(x^n\right)=n\times x^{n-1}$ then we will use ${\displaystyle \frac{d}{dx}}(\sin x)=\cos x$ .

Complete step by step answer:
We have been given a function ${\sin }^2\left(2x\right)$ .
We have to find the derivative of the given function.
Now let us assume that $y={\sin }^2\left(2x\right)$
Now, differentiating the given function with respect to x we will get
 $\Rightarrow {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{d}{dx}}\left({\sin }^2\left(2x\right)\right)$
Now, the given function is a composite function so we need to apply the chain rule.
Now, we know that ${\displaystyle \frac{d}{dx}}\left(x^n\right)=n\times x^{n-1}$
Applying the formula in the above equation we will get
 $\Rightarrow {\displaystyle \frac{dy}{dx}}=2\sin 2x{\displaystyle \frac{d}{dx}}(\sin \left(2x\right))$
Now, we know that ${\displaystyle \frac{d}{dx}}(\sin x)=\cos x$
Applying the formula in the above-obtained equation we will get
 $\Rightarrow {\displaystyle \frac{dy}{dx}}=2\sin 2x.\cos 2x{\displaystyle \frac{d}{dx}}2x$
Taking the constant term out we will get
 $\Rightarrow {\displaystyle \frac{dy}{dx}}=2\sin 2x.\cos 2x\times 2{\displaystyle \frac{d}{dx}}x$
Now, we know that ${\displaystyle \frac{d}{dx}}x=1$
Applying the formula in the above-obtained equation we will get
 $\Rightarrow {\displaystyle \frac{dy}{dx}}=2\sin 2x.\cos 2x\times 2$
Now, solving further we will get
 $\Rightarrow {\displaystyle \frac{dy}{dx}}=4\sin 2x.\cos 2x$
Hence we get the derivative of ${\sin }^2\left(2x\right)$ as $4\sin 2x.\cos 2x$ .

Note:
We can also further simplify the obtained differentiation value by using trigonometric identities and formulas. Also do not get confused between the product rule and chain rule of the differentiation. The product rule is applied when one function is multiplied by another function. We cannot directly differentiate the product of functions or a composite function.