Question

How do you differentiate \[f(x)=\sin \left( {{x}^{3}} \right)\]?

Answer 1

Hint: This problem is from the chapter of derivation. For solving this question, we are going to use some formulas of derivatives like \[\dfrac{d}{dx}\sin x=\cos x\] and \[\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}\]. After the first step of differentiation, we will have to use chain rule in solving the question to get the exact differentiation. The chain rule helps us to differentiate composite functions. And, \[\sin \left( {{x}^{3}} \right)\] is a composite function because it is in the form of f(g(x)) where g(x) is the function of x and f(x) is function of sin.

Complete step by step answer:
Let us solve this question.
In this question, it is asked to differentiate the function\[f(x)=\sin \left( {{x}^{3}} \right)\].
So, the differentiation of \[f(x)=\sin \left( {{x}^{3}} \right)\] will be
\[\dfrac{d}{dx}f(x)=\dfrac{d}{dx}\sin \left( {{x}^{3}} \right)\]
By using the formula \[\dfrac{d}{dx}\sin x=\cos x\] in the above equation, we get
\[\dfrac{d}{dx}f(x)=\cos \left( {{x}^{3}} \right)\dfrac{d}{dx}\left( {{x}^{3}} \right)\]
Here, we have applied the chain rule.
Let us first know that what chain rule is. The chain rule states that the derivative of \[f\left( g\left( x \right) \right)\] is \[f'\left( g\left( x \right) \right)\times g'\left( x \right)\]. The chain rule helps us to differentiate composite functions. \[\sin \left( {{x}^{3}} \right)\] is a composite function as it is in the form of f(g(x)) where g(x) is the function of x and f(x) is function of sin.
Now, coming to the equation of differentiation.
\[\Rightarrow \dfrac{d}{dx}f(x)=\cos \left( {{x}^{3}} \right)\dfrac{d}{dx}\left( {{x}^{3}} \right)\]
Applying the formula \[\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}\] in the above differentiation, we get
\[\Rightarrow \dfrac{d}{dx}f(x)=\cos \left( {{x}^{3}} \right)\times 3{{x}^{2}}\]
\[\Rightarrow \dfrac{d}{dx}f(x)=\cos \left( {{x}^{3}} \right)\times 3{{x}^{2}}=3{{x}^{2}}\cos \left( {{x}^{3}} \right)\]
Hence, the differentiation of \[f(x)=\sin \left( {{x}^{3}} \right)\] is \[3{{x}^{2}}\cos \left( {{x}^{3}} \right)\].

Note:
For solving this type of problem, we should have a better knowledge of differentiation.
Some formulas should be kept remembered to make the solution process easier.
The formulas are:
\[\dfrac{d}{dx}\sin x=\cos x\] ,
\[\dfrac{d}{dx}\cos x=-\sin x\], and
\[\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}\]
We should not forget to apply the chain rule. Otherwise, the solution will be wrong in this type of question. And, always remember that chain rule is used to differentiate composite functions.