Question

How do you find the derivative of $y = {e^{5x}}$ ?

Answer 1

Here the basic concept which we are going to use is using the chain rule. We will find the derivative with respect to x using the chain rule. We have to apply the chain rule here because there is some numerical value other than 1 in place of the coefficient of x.

Complete Step by Step Solution:
The given equation is $y = {e^{5x}}$
Differentiating both sides with respect to x, we get
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( {{e}^{5x}} \right)}{dx}$
As we know the first derivative of ${e^x}$ is ${e^x}$, but here in the equation, there is 5, present in place of the coefficient of x, so we have to apply the chain rule to find its derivative.
The chain rule states that the derivative of $f\left( {g\left( x \right)} \right)$ is $f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)$ , helps us differentiate composite functions.
So, here first we will differentiate ${e^{5x}}$ with respect to x, and then we will differentiate $\;5x$ with respect to x.
$\Rightarrow \dfrac{{dy}}{{dx}} = {e^{5x}} \cdot \dfrac{{d\left( {5x} \right)}}{{dx}}$
$\Rightarrow \dfrac{{dy}}{{dx}} = {e^{5x}} \cdot 5$
Rewriting above equation

$\Rightarrow \dfrac{{dy}}{{dx}} = 5{e^{5x}}$

Additional Information:
The chain rule is very important as we have to use it in many of the problems related to finding derivatives of composite functions.

Note:
There is an alternative method to solve this by taking log on both sides
Given equation $y = {e^{5x}}$
Take log both sides
$\Rightarrow {\log _e}y = {\log _e}{e^{5x}}$
$\Rightarrow {\log _e}y = 5x \cdot {\log _e}e$
As we know that ${\log _e}e = 1$
Hence, ${\log _e}y = 5x$
We know that $\dfrac{{d\left( {{{\log }_e}x} \right)}}{{dx}} = \dfrac{1}{x}$
So, now differentiating both sides with respect to x,
$\Rightarrow \dfrac{1}{y}\left( {\dfrac{{dy}}{{dx}}} \right) = 5$
$\Rightarrow \dfrac{{dy}}{{dx}} = 5y$
As given in the question, $y = {e^{5x}}$ , therefore
$\Rightarrow \dfrac{{dy}}{{dx}} = 5{e^{5x}}$.