Derivative of e^-x

Learn what is the derivative of e^-x with formula and proof. Also understand how to prove the derivative of e^-x by first principle of differentiation.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of e^-x

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 e^-9 by applying the first principle of differentiation. In this article, you will learn what the derivative of e-x is and how to calculate the derivative of e-x by using different approaches.

What is the derivative of e^-x?

The derivative of the exponential function e-x with respect to x is given by d/dx (e^-x) = -e^-x This represents the instantaneous rate of change of the function at any point along its curve.

Interestingly, the differentiation of e^-x is equal to its inverse, -e^(-x), which is always negative. This derivative is commonly used in calculus, physics, and engineering to model exponential decay or growth

Derivative of e^-x formula

The formula of derivative of e to the -x is equal to exponential function e, that is;

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

How do you prove the derivative of e-x?

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

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

Derivative of e-x by first principle

The derivative of a function by 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,

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

Proof of derivative of e-x by first principle

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


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


$f'(x) = \lim_{h\to 0}\frac{e^{-x}.e^{-h} - e^{-x}}{h}$

Taking e-x common as;

$f'(x) = \lim_{h\to 0}\frac{e^{-x}(e^{-h} - 1)}{h}$

More simplification,

$f'(x) = -e^{-x}.\lim_{h\to 0}\frac{(e^{-h} - 1)}{-h}$

When h approaches to zero,

$f'(x) = -e^{-x}\lim_{h\to 0}\frac{(e^0 - 1)}{-h}$

$f'(x) = -e^{-x}f(0)$


$f'(x) = -e^{-x}$

Hence the derivative of e-x is the same as the derivative of ex with a minor difference of negative sign.

Derivative of e^-x by product rule

The derivative of e can be calculated by using product rule because the cosine function can be written as the combination of two functions. The product rule of derivatives is defined as;

$\frac{d}{dx}[uv] = \frac{du}{dx}.v +u.\frac{dv}{dx}$

Proof of derivative of e^(-x) by product rule

To prove the derivative of e by using product rule, assume that,

$f(x) = 1. e^{-x}$

By using product rule of differentiation,

$f'(x) = \frac{d}{dx}(e^{-x}). 1 + e^{-1}\frac{d}{dx}(1)$

We get,

$f'(x) = -e^{-x} + 0$


$f'(x) = - e^{-x}$

Derivative of e-x using quotient rule

Since the cotangent is the reciprocal of tangent. Therefore, the derivative of cot can also be calculated by using the quotient rule. The quotient rule is defined as;

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

Proof of derivative of e-x by quotient rule

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

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

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

$\frac{d}{dx}(\frac{u(x)}{v(x)}) = \frac{u(x). v'(x) -v(x).u'(x)}{(v(x))^2}$

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

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

$f'(x)= -e^{-x}$

Hence, we have derived the derivative e^{-x} using the quotient rule of differentiation.

How to find the derivative of e-x 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 e^-x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the differential of e^-x.
  2. Now, select the variable by which you want to differentiate e-x. Here you have to choose x.
  3. Select how many times you want to differentiate e-x 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 e-x within a few seconds.

Frequently asked questions

What are the types of exponential functions?

Exponential functions are classified into two types: exponential growth and exponential decay. When b > 1, the function f (x) = bx represents exponential growth. When 0 b 1 and f (x) = bx, the function represents exponential decay.

What is the derivative of e^-x?

The derivative of e-x with respect to x is -e-x. Mathematically, the derivative of e to the x is written as;

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

Related Problems

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