# 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.eh - 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)

Therefore,

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)

Hence,

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.