Question

How to find the derivative of $Sin3{x^2}$ ?

Answer 1

Hint: Derivative is the process of finding small changes in function with respect to the given variable. There are basic formulae with which we can find derivatives like $\dfrac{d}{{dx}}\operatorname{Sin} x = \operatorname{Cos} x$ .
But, in a given problem there is an implicit function. These types of problems can be solved using chain derivative methods.

Complete step-by-step answer:
Given function : \[y = 3{x^2}\]
We can look at the function as \[Siny = Sin3{x^2}\]
Where, \[y = 3{x^2}\]
   \[ \Rightarrow \dfrac{d}{{dy}}\left( {Sin3{x^2}} \right) = Cos3{x^2} \times \dfrac{d}{{dx}}(3{x^2})\]
Here as stated above \[y = 3{x^2}\]
  \[ \Rightarrow \dfrac{d}{{dy}}\left( {Sin3{x^2}} \right) = Cos3{x^2} \times \dfrac{d}{{dx}}(3{x^2})\] ………. \[\left( 1 \right)\]
As we know, \[ \Rightarrow \dfrac{d}{{dx}}(a{x^2}) = 2ax\] …… $a$ is constant
  \[ \Rightarrow \dfrac{d}{{dy}}\left( {Sin3{x^2}} \right) = Cos3{x^2} \times 2 \times 3 \times x\]
  \[ \Rightarrow \dfrac{d}{{dy}}\left( {Sin3{x^2}} \right) = {\text{6}}xCos(3{x^2})\]
Answer is \[ \to {\text{6}}xCos(3{x^2})\]
So, the correct answer is “ \[ \to {\text{6}}xCos(3{x^2})\] ”.

Note: This method can be applied on a range of problems involving implicit functions. Also, there can be multiple implicit functions in a given function for example $Cos(3sin(4{x^3}))$ . In such cases the derivatives of the implicit functions are kept on multiplying until $\dfrac{{dx}}{{dx}}$i.e. $1$ .
Assuming the inner function \[f\left( y \right)\] and taking derivatives with respect to helps in avoiding mistakes.