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.