Question

What is the derivative of \[{{\sin }^{2}}\left( 4x \right)\]?

Answer 1

Hint: For solving this question you should know about the differentiation of trigonometric functions and we can solve this by this. Here, we can see that two terms are looking so we will differentiate to outside terms first and then we differentiate the inside term. Thus, we will get the answer for this.

Complete step by step solution:
According to our question we have to calculate the derivative of \[{{\sin }^{2}}\left( 4x \right)\].
As we know that the differentiation of trigonometric functions will always give us trigonometric values and if we differentiate to inverse trigonometric functions then it will give any other value than trigonometric functions.
And by the differentiation of any variable it will reduce the power of the variable by 1 and the power will multiply by it again.
So, according to our question, \[y={{\sin }^{2}}\left( 4x \right)\].
So, if we consider, \[y={{\left( f\left( x \right) \right)}^{2}}={{\left( \sin \left( 4x \right) \right)}^{2}}\].
Then if we differentiate it,
\[y={{\left( f\left( x \right) \right)}^{n}}\Rightarrow \dfrac{dy}{dx}=n.f'\left( x \right).{{\left( f\left( x \right) \right)}^{n-1}}\]
So, according to this equation we can write, \[y={{\left( \sin \left( 4x \right) \right)}^{2}}\]
If we differentiate it, then
\[\dfrac{dy}{dx}=2.\sin \left( 4x \right).\dfrac{d}{dx}\sin \left( 4x \right)\]
Now, solve the differentiation of \[\sin \left( 4x \right)\].
\[\dfrac{dy}{dx}=2.\sin \left( 4x \right).\cos \left( 4x \right).\dfrac{d}{dx}\left( 4x \right)\]
Since, we know that, \[\dfrac{d}{dx}\sin x=\cos x\].
If we differentiate the 4x by with respect to x then,
\[\dfrac{dy}{dx}=2.\sin \left( 4x \right).\cos \left( 4x \right).4\]
We can write it as,
\[\dfrac{dy}{dx}=8.\sin \left( 4x \right).\cos \left( 4x \right)\]
So, the derivative of \[{{\sin }^{2}}\left( 4x \right)\] is \[8.\sin \left( 4x \right).\cos \left( 4x \right)\].

Note: For calculating the differentiation of any trigonometric function we have to learn all the trigonometric formulas for differentiation because this is completely based on the formulas. And always we should be careful of the power of variables and especially if they are the variable whose regarding we are differentiating that.