Derivative of e^f(x)

Learn what is the derivative of e^f(x) with easy and step-wise proof. Also understand to prove the derivative of e^f(x) by product and quotient rule.

Alan Walker-

Published on 2023-09-04

Introduction to the Derivative of e^f(x)

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of e^f(x) can be calculated by following the rules of differentiation.

Or, we can directly find the e to the x derivative by applying the first principle of differentiation. In this article, you will learn what the e^f(x) derivative formula is and how to calculate the derivative e^f(x) by using different approaches.

What is the derivative of e^f(x)?

The derivative of e^(f(x)) with respect to x is f’(x)e^f(x), represented by d/dx(e^f(x)). This formula expresses the rate of change of the exponential function e concerning x. As a fundamental rule of calculus, the derivative of e^f(x) always equals the exponential function itself. This property makes it a crucial tool in solving various mathematical problems. 

Derivative formula of e^f(x)

The derivative formula of e^f(x) states that the derivative of e to the f(x) with respect to x is equal to the exponential function f’(x)e^f(x). This is expressed mathematically as;

d / dx (ef(x)) = f’(x).ef(x)

Understanding the derivative formula and its practical applications can help to determine the behavior of exponential functions and their derivatives in physics, economics, engineering, and other disciplines.

How do you prove the derivative e^f(x)?

There are various ways to prove the e^f(x) derivative formula. These are;

  1. First Principle

  2. Product Rule

  3. Quotient Rule

Each method provides a different way to compute the exponential derivative. By using these methods, we can mathematically prove the formula for finding the differential of e^f(x).

Derivative of e^f(x) by first principle

A fundamental way to find the derivative of a function is by using the first principle, which is also known as the delta method. This method involves finding a general expression for the slope of a curve by using algebra. The derivative is a measure of the instantaneous rate of change of a function at a specific point, and it can be calculated using the limit formula:

f(x) = lim f(x + h) - f(x) / h

We can use this formula to prove the exponential derivative formula for e^f(x).

Proof of e to the f(x) derivative by first principle

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

dy/dx = limh→0f(x + h) - f(x) / h

dy/dx = lim ef(x+h) - ef(x)/h

Moreover, we can replace f(x) by eu to calculate the derivative of e^u..

f(x) = lim - ex/h

Taking ex common as;

f(x) = lim ef(x)(ef(x+h)-f(x) - 1) / h

More simplification,

f(x) = ef(x) .lim (ef(x+h)-f(x) - 1) / h

When h approaches to zero,

f(x) = ef(x) lim (e0 - 1) / h

f(x) = ef(x) f’(x)


f(x) = f’(x)ex

Hence we have verified the derivative of ef(x) and this method can be used to calculate the derivative of any exponential function.

Derivative of e to the f(x) by product rule

The product rule in derivatives is used when we have to calculate the derivative of two functions at a time. The product rule formula is;

[uv] = u.v + u.v

The derivative of e^f(x) can be calculated by using the product rule formula because the function e^x can be written as the combination of two functions. 

Proof of derivative of e by product rule

To prove the derivative e f(x) by using the product rule, we start by assuming that,

y = 1. ef(x)

By using the product rule of differentiation,

dy/dx = (1). ef(x) + (ef(x))’

We get,

dy/dx = 0 + f’(x)ef(x)


dy/dx = f’(x) ef(x)

We can also verify the derivative of e^f(x) by using the product rule calculator.

Derivative of e to the f(x) using quotient rule

Another method for finding the derivative of e^f(x) is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of the exponential function e^x can also be calculated using the quotient rule. The quotient rule is defined as:

d / dx (f/g) = f(x). g(x) - g(x).f(x) /{g(x)}2

Proof of derivative of e by quotient rule

To prove the e to the f(x) derivative, we start by writing it as,

f(x) = ef(x) /1 = u/v

Supposing that u = ef(x) and v = 1. Now by the quotient rule,

f(x) = (vu - uv)/v2

f(x) = [ef(x) d / dx(1) - 1. d / dx(ef(x))] / (1)2

= [ef(x) (0) - 1 (f’(x)ef(x))] / 1

= f’(x)ef(x) / 1

= f’(x)ef(x)

Hence, the derivative of e has been derived. Or, you can also use a quotient calculator.

How to find the e^f(x) derivative 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 ef(x) 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 e.

  2. Now, select the variable by which you want to differentiate e^f(x). Here you have to choose x.

  3. Select how many times you want to differentiate e to the f(x). In this step, you can choose 2 to calculate the second derivative, 3 for the third derivative and so on.

  4. Click on the calculate button. After this step, you will get the derivative of e^f(x) within a few seconds.

After completing these steps, you will receive the e derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

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