Question

What is the derivative of ${e^{2{x^2}}}$?

Answer 1

Hint : Here, the given question has a trigonometric function. We have to find the derivative or differentiated term of the function. First consider the function $y$, then differentiate $y$ with respect to $x$ by using a standard differentiation formula of trigonometric ratio and use chain rule for differentiation. And on further simplification we get the required differentiate value.

Complete step-by-step answer :
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.
Consider the given function
$y = {e^{2{x^2}}} - - - - \left( 1 \right)$
We know the chain rule, that is $y = {e^{g(x)}}$ then the derivative is given by
${y^1} = {e^{g(x)}}g'\left( x \right)$.
Here $g\left( x \right) = 2{x^2}$.
now differentiating (1) with respect to x
$\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{e^{2{x^2}}}} \right)$
$\dfrac{{dy}}{{dx}} = {e^{2{x^2}}}\dfrac{d}{{dx}}\left( {2{x^2}} \right)$.
We know the differentiation of ${x^n}$ is $\dfrac{{d({x^n})}}{{dx}} = n.{x^{n - 1}}$.
$\dfrac{{dy}}{{dx}} = {e^{2{x^2}}}\left( {2.2{x^{2 - 1}}} \right)$
$\dfrac{{dy}}{{dx}} = {e^{2{x^2}}}\left( {4x} \right)$
$\dfrac{{dy}}{{dx}} = 4x{e^{2{x^2}}}$
Thus the derivative of ${e^{2{x^2}}}$ with respect to x is $4x{e^{2{x^2}}}$.
So, the correct answer is “ $4x{e^{2{x^2}}}$”.

Note : We know the differentiation of ${x^n}$ is $\dfrac{{d({x^n})}}{{dx}} = n.{x^{n - 1}}$. The obtained result is the first derivative. If we differentiate again we get a second derivative. If we differentiate the second derivative again we get a third derivative and so on. Careful in applying the product rule. We also know that differentiation of constant terms is zero