Question

If \[y = {\left( {\cos {x^2}} \right)^2}\], then \[\dfrac{{dy}}{{dx}}\] is equal to
1) \[ - 4x\sin 2{x^2}\]
2) \[ - x\sin {x^2}\]
3) \[ - 2x\sin 2{x^2}\]
4) \[ - x\sin 2{x^2}\]

Answer 1

Hint: Here, the given question. We have to find the derivative or differentiated term of the given trigonometric function. For this, first consider the function \[y\], then differentiate \[y\] with respect to \[x\] by using standard differentiation formulas of trigonometric functions and using chain rule for differentiation. And on further simplification we get the required differentiation value.

Complete step-by-step answer:
Differentiation can be defined as a derivative of a function with respect to an independent variable
Otherwise
The differentiation of a function is defined as the derivative or rate of change of a function. The function is said to be differentiable if the limit exists.
Let \[y = f\left( x \right)\] be a function of. Then, the rate of change of “y” per unit change in “x” is given by \[\dfrac{{dy}}{{dx}}\].
The Chain Rule is a formula for computing the derivative of the composition of two or more functions.
The chain rule expressed as \[\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \cdot \dfrac{{du}}{{dx}}\]
Consider the given function \[y\]
 \[ \Rightarrow y = {\left( {\cos {x^2}} \right)^2}\]---------- (1)
Here, \[y\] is a dependent variable and \[x\] is an independent variable.
Now, differentiate function \[y\] with respect to \[x\]
\[ \Rightarrow \dfrac{d}{{dx}}\left( y \right) = \dfrac{d}{{dx}}\left( {{{\left( {\cos {x^2}} \right)}^2}} \right)\]
On differentiating using a formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = {x^{n - 1}}\], then by chain rule we have
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2\cos {x^2}\dfrac{d}{{dx}}\left( {\cos {x^2}} \right)\]
On differentiating using a formula \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\], and then
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2\cos {x^2}\left( { - \sin {x^2}} \right)\dfrac{d}{{dx}}\left( {{x^2}} \right)\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2\cos {x^2}\left( { - \sin {x^2}} \right)\left( {2x} \right)\]
On simplification, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = - 2x \cdot 2\cos {x^2}\sin {x^2}\]
Apply a double angle formula of trigonometry i.e., \[\sin 2x = 2\sin x\cos x\], then we get
\[\therefore \dfrac{{dy}}{{dx}} = - 2x\sin 2{x^2}\]
Hence, it’s a required solution.
Therefore, option (3) is the correct answer.
So, the correct answer is “Option 3”.

Note: The student must know about the differentiation formulas for the trigonometric, algebraic, functions and these differentiation formulas are standard. If the function is complex, we have to use the chain rule differentiation. It makes it easy to find out the differentiated term means to differentiate a big function step by step.