Derivative of \[e^{2x}\] ?

Hint: In this question , we need to find the derivative of \[e^{2x}\] . Mathematically, a derivative is defined as a rate of change of function with respect to an independent variable given in the function. The term differentiation is nothing but it is a process of determining the derivative of a function at any point. With the help of the derivative chain rule, we can find the derivative of \[e^{2x}\].
Chain rule :
Chain rule is the derivative of the composite function which is the product of the derivative of the first function and derivative of the second function of the composite function. The use of chain rule is to find the derivative of the composite function.
\[\dfrac{dy}{{dx}} = \dfrac{{du}}{{dx}} \times \dfrac{{dy}}{{du}}\]
Where, \[\dfrac{{dy}}{{dx}}\ \] is the derivative of \[y\] with respect to \[x\]
\[\dfrac{{du}}{{dx}}\ \] is the derivative of u with respect to \[x\]
\[\dfrac{{dy}}{{du}}\] is the derivative of \[y\] with respect to \[u\]
Derivative formulae used :
1. \[\dfrac{d}{{dx}}\left( x \right) = 1\]
2. \[\dfrac{d}{{dx}}\left( e^{x} \right) = e^{x}\]

Complete step by step solution:
Given, \[e^{2x}\]
Let us consider \[y = e^{2x}\ \]
Here we will chain rule to find the derivative of \[e^{2x}\] .
Chain rule :
\[\dfrac{dy}{{dx}} = \dfrac{{du}}{{dx}} \times \dfrac{{dy}}{{du}}\]
\[\Rightarrow y = e^{2x}\]
Let us consider \[2x = u\]
\[\Rightarrow u = 2x\]
We get, \[y = e^{u}\]
First we can differentiate \[u\],
\[u = 2x\]
Differentiating both sides with respect to \[x\] ,
We get,
\[\Rightarrow\dfrac{d}{{dx}}\left( u \right) = \dfrac{d}{{dx}}\left( 2x \right)\]
By taking the constant outside,
\[\Rightarrow\dfrac{d}{{dx}}\left( u \right) = 2\dfrac{d}{{dx}}\left( x \right)\]
We know that \[\dfrac{d}{{dx}}\left( x \right) = 1\]
Thus we get, \[\dfrac{{du}}{{dx}} = 2\]
Now we need to differentiate \[y = e^{u}\]
Differentiating both sides with respect to \[u\],
\[\Rightarrow\dfrac{d}{{du}}\left( y \right) = \dfrac{d\left( e^{u} \right)}{{du}}\]
We know that \[\dfrac{d}{{dx}}\left( e^{x} \right) = e^{x}\]
Thus we get, \[\dfrac{{dy}}{{du}} = e^{u}\]
By substituting the values in the chain rule formula,
\[\Rightarrow\dfrac{dy}{{dx}} = \dfrac{{du}}{{dx}} \times \dfrac{{dy}}{{du}}\]
Here, \[\dfrac{{du}}{{dx}} = 2\] and \[\dfrac{{dy}}{{du}} = e^{u}\]
We get,
\[\Rightarrow\dfrac{{dy}}{{dx}} = 2 \times e^{u}\]
By substituting \[u = 2x\] ,
We get,
\[\dfrac{dy}{{dx}} = 2e^{2x}\]
Thus we get the derivative of \[e^{2x}\] is \[2e^{2x}\]
The derivative of \[e^{2x}\] is \[2e^{2x}\]

Note: Mathematically, derivative helps in solving the problems in calculus and in differential equations. The derivative of \[y\] with respect to \[x\] is represented as \[\dfrac{{dy}}{{dx}}\]. Here the notation \[\dfrac{{dy}}{{dx}}\] is known as Leibniz's notation . In derivation, there are two types of derivative namely first order derivative and second order derivative. A simple example for a derivative is the derivative of \[x^{3}\] is \[3x\] . Derivative is applicable in trigonometric functions also.