Question

How do you differentiate \[y = \sin {x^2}\]?

Answer 1

Hint: Here, we have to find the derivative of the given function. We will differentiate the given function by using the chain rule and by using the derivative formula. Differentiation is a method of finding the derivative of the function and finding the rate of change of a function with respect to one variable.

Formula used:
We will use the following formulas:
Derivative formula: \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\]
Derivative formula of Power rule: \[\dfrac{d}{{dx}}\left( {{x^2}} \right) = 2x\]

Complete Step by Step Solution:
We are given a function \[y = \sin {x^2}\].
Now, by differentiating both sides of the given function with respect to \[x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\sin {x^2}} \right)\]
We know that if a function has two functions, then both the functions have to be differentiated separately.
We are given a trigonometric function of the sine of the algebraic function raised to the power 2. First, we will find the derivative of the trigonometric function and then the derivative of the algebraic function.
By using the derivative formula \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\] and by using the chain rule, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \cos {x^2} \cdot \dfrac{d}{{dx}}\left( {{x^2}} \right)\]
Now using the derivative formula of Power rule \[\dfrac{d}{{dx}}\left( {{x^2}} \right) = 2x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \cos {x^2} \cdot 2x\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 2x.\cos {x^2}\]

Therefore, the derivative of \[y = \sin {x^2}\] is\[2x\cos {x^2}\].

Note: We have used the chain rule to find the derivative of the composite functions. Chain rule states that if a composite function \[f\left[ {g\left( x \right)} \right]\] where \[f\left( x \right)\] and \[g\left( x \right)\] be two functions, then the derivative of the composite function is\[f'\left[ {g\left( x \right)} \right] \cdot g'\left( x \right)\]. If there is an arithmetic operation of addition and subtraction, then it appears as the same, but if no such arithmetic operation exists, then all these functions have to be the product of all the functions.