Question

How do you differentiate ${\sin ^2}\left( {2x} \right)$ ?

Answer 1

Hint: Use the chain rule to do the differentiation of the function in the question and use the formula of differentiation, ${x^n} = n{x^{n - 1}}$. Now, find the way to do the differentiation of the sine function. In chain rule, each function is differentiated one by one and then multiplied with each other.

Complete Step by Step Solution:
Differentiation and integration are the two mathematical concepts which when composed form the calculus and both are opposite of each other. Differentiation can be defined as an instantaneous rate of change of the function with respect to one of its variables. It is the change in the value of function due to change in the independent variable. If a function $y$ is differentiated with respect to $x$ , then any change in the value of $y$ due to the change in the value of $x$ is given by –
$\dfrac{{dy}}{{dx}}$
In the question, we have been given the function ${\sin ^2}\left( {2x} \right)$. So, the function can also be written as-
$ \Rightarrow {\left( {\sin \left( {2x} \right)} \right)^2}$
As there is the power of 2 in the above function. Therefore, the formula to differentiate any function having power is –
$\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$, where, $x$ is the function and $n$ is the power of that function.
We also know that the formula to differentiate $\sin x$ which is –
$ \Rightarrow \dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x$
So, let $y = {\left( {\sin \left( {2x} \right)} \right)^2}$, hence, now differentiating $y$ with respect to $x$ -
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}{\left( {\sin (2x)} \right)^2}$
Using the formulas of differentiation as discussed above, we get –
$
   \Rightarrow \dfrac{{dy}}{{dx}} = 2\sin 2x \times \dfrac{d}{{dx}}\left( {\sin 2x} \right) \\
   \Rightarrow \dfrac{{dy}}{{dx}} = 2\sin 2x\cos 2x \times 2 \\
   \Rightarrow \dfrac{{dy}}{{dx}} = 4\sin 2x\cos 2x \\
 $

Hence, the differentiation of the function ${\sin ^2}\left( {2x} \right)$ is $4\sin 2x\cos 2x$.

Note:
The integration is the opposite of the differentiation, so, if we integrate the function $4\sin 2x\cos 2x$ with respect to $x$ we will again get back the function which is written in the question, ${\sin ^2}\left( {2x} \right)$. We usually do the integration by using the substitution method.