Derivative of e^5x

Learn what is the derivative of an exponential function e to the 5x with formula. Also understand how to prove the derivative of e^5x by first principle.

Alan Walker-

Published on 2023-07-12

Introduction to the derivative of e^5x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of e can be calculated by following the rules of differentiation. Or, we can directly find the derivative of e5 by applying the first principle of differentiation. In this article, you will learn what the derivative of e5 is and how to calculate the derivative of e5x by using different approaches. 

What is the derivative of e^5x?

The derivative of e^5x is equal to 5e^5x. This can be represented as d/dx (e^5x). Essentially, the differential of e^5x measures the rate of change of the function; in this case, it is always equal to the negative of the original function. Understanding the derivative of e^(5x) is important in various fields of mathematics and sciences, such as calculus and physics.

Derivative of e5x formula

The formula for the differentiation of e^(5x) is equal to the 5 multiplied by e^(5x). Mathematically,

 $\frac{d}{dx}(e^{5x}) = 5e^{5x}$

It is important to note that the derivative of e^(5x) is not the same as the derivative of e^(x), which is equal to e^(x).

How do you prove the derivative of e5x?

 There are various ways to derive derivatives of e^5x. Therefore, we can prove the derivative of e^5x by using;

  1. First Principle
  2. Product Rule
  3. Quotient Rule

Derivative of e5x by first principle

According to the first principle of derivative, the ln e^(5x) derivative is equal to 5e^(5x). The derivative of a function by the first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of the derivative.

Proof of derivative of e5x by first principle

To prove the derivative of e^5x by using first principle, replace f(x) by e^5x.


$f’(x) =\lim_{h\to 0}\frac{e^{5(x+h)} – e^{5x}}{h}$


$f’(x) =\lim_{h\to 0}\frac{e^{5x}.e^{5h} – e^{5x}}{h}$

Taking e^5x common as;

$f’(x) =\im_{h\to 0}\frac{e^{5x}(e^{5h} – 1)}{h}$

More simplification, 

$f’(x) = 5e^{5x}.\lim_{h\to 0}\frac{(e^{5h} – 1)}{5h}$

When h approaches to zero, 

$f’(x) = 5e^{5x}\lim_{h\to 0}\frac{(e^0 – 1)}{5h}$

$f’(x) = 5e^{5x} f’(0)$


$f’(x) = 5e^{5x}$

Also, we can use the derivative by definition calculator to find rate of change of a function without any manual calculations.

Derivative of e^5x by product rule

Another method to find the derivative e^(4x) is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule is:

$\frac{d}{dx}(uv) = u\frac{dv}{dx} + \frac{du}{dx}v{2}nbsp;

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of e^5x by product rule 

To prove the derivative of e^5x by using product rule, we start by assuming that,

$f(x) = e^{2x}. e^{3x}$

By using product rule of differentiation,

$f’(x) = (e^{2x})’. e^{3x} + (e^{3x})’e^{2x}$

We get,

$f’(x) = 2e^{5x} + 3e^{5x}$


$f’(x) = 5e^{5x}$

The above calculations can also be done by using the product rule calculator with steps.

Derivative of e4x using quotient rule

Another way of finding the derivative of e^5x is the quotient rule formula. This method is used when we have to deal with a fraction of two functions. The quotient rule is defined as;

$\frac{d}{dx}(\frac{f}{g}) = \frac{f(x). g’(x) –g(x).f’(x)}{(g(x))^2}$

Proof of derivative of e5x by quotient rule 

To prove the derivative of e^5x, we can write it,

$f(x) = \frac{e^{5x}}{1} = \frac{u}{v}$

Supposing that u = e^5x and v = 1. Now by quotient rule,

$f’(x) = \frac{(vu’ – uv’)}{v^2}$

$f'(x) = \frac{\frac{d}{dx}(e^{5x}) – e^{5x}.\frac{d}{dx}(1)}{(1)^2}$

$f'(x)= \frac{5e^{5x}}{1}$

$f'(x)= 5e^{5x}{2}nbsp;

Hence, we have derived the derivative of e5x using the quotient rule of differentiation calculator.

How to find the derivative of e5x with a calculator?

The easiest way to calculate the derivative of e is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as e5x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of e5x.
  2. Now, select the variable by which you want to differentiate e5x. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate e5x to the x. In this step, you can choose 2 for second, 3 for third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of e5x within a few seconds.


What is the derivative of e^5x?

The derivative of e5x with respect to x is 5e^5x. Mathematically, the derivative of e squared to the x is written as;

$\frac{d}{dx} (e^{5x}) = 5e^{5x}$

How do you use the derivative rule for exponential?

The derivative of a power is equal to the power itself multiplied by the following: the exponent's derivative times the logarithm of the base, plus the base's derivative times the exponent-base ratio.

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