Derivative of e^x is e^x. Derivative of e^x means finding the change in the exponential function with respect to the independent variable. The process of finding the derivative is known as differentiation. The derivative of e^x is e^x. Understanding the derivative of e^x is an important concept in calculus as it offers insights into the dynamic nature of exponential growth.

In this article, we will talk about derivatives of e^x, what is derivative, and its some basic rules.

What is the Derivative of e^x?

The derivative of e^x means the change in the exponential with respect to x. It is denoted by d(e^x)/ dx. It is written as f(x) = e^x, where ‘e’ is the Euler’s number and its value is approximately 2.718. The derivative of e^x is e^x

Derivative of e^x Formula

Formula for the derivative of e^x is given below,

d(e^x)/dy = e^x

(e^x)’ = e^x

Learn More:

• Calculus in Maths
• Derivative in Math
• Differential Calculus

Proof of Derivative of e^x

The derivative of cot x can be proved using the following ways:

• By using First Principle of Derivative
• By using Derivative of a^x

Differentiation of e^x Using First Principle

We are going to prove the differentiation of e^x is e^x by using the first principle of derivatives. We will use some basic rules and formulas of exponential functions and derivatives that are given below.

f'(x) = lim h→0 [f(x + h) – f(x)] / h

Also, we know that,

e^x+h = e^x.e^h

lim _x→0 (e^x – 1) / x = 1

Using above formulas, we get,

d(e^x)/dx = lim _h→0 [e^x+h – e^x] / h

= lim _h→0 [e^x.e^h – e^x] / h

= lim _h→0 e^x [e^h – 1] / h

= e^x lim _h→0 [e^h – 1] / h

= e^x × 1 = e^x

Hence we have proved that the derivative of e^x to be equal to e^x

Differentiation of e^x Using Derivative of a^x

Exponential function are expressed in the form f(x) = a^x , where ‘a’ is a constant (real number) and x is variable.

Derivative of exponential function f(x) = a^x is find by the formula,

f'(x) = (ln a).a^x…(i)

Substituting value a = e in eq (i), we get the differentiation of e^x which is given by

f'(x) = (ln e) e x

= 1 × e^x [As, ln e = 1]

= e^x

Hence, the derivative of e to the power x is e x .

People Also Read:

• Differentiation of Trigonometric Function
• Differentiation Formulas
• Derivative of e^2x 

Solved Examples on Derivative of e^x

Some examples related to Derivative of e^x are,

Example 1: Find the derivative of e^2x.

Solution:

Using Chain Rule

y = e^2x

y’ = d(e^2x)/dx

y’ = [d(e^2x)/dx ].[ d(2x)/dx]

y’ = e^2x. 2

y’ = 2(e^2x)

Example 2:Find the derivative of e^x/x².

Solution:

Using Quotient Rule

[(d[u/v]/dx) = [u’v – uv’]/v²]

y = e^x/x²

y’ = d(e^x/x²)/dx

y’ = ( d(e^x)/dx . x² – e^x. d(x²)/dx )/ (x²)²

y’ = ( e^x. x² – e^x. 2x )/ x⁴

y’= ( x .e^x( x – 2))/x⁴

y’ = ( e^x. (x-2) )/ x³

Example 3: Find the derivative of the function (e^x)^x

Solution:

y= (e^x)^x

Taking log both side, we get,

ln y = ln (e^x)^x

ln y = x.(ln e^x )

ln y = x²

Differentiation both side, we get,

1/y . y’ = 2x

y’ = y. (2x)

y’ = (e^x)^x. (2x)

Example 4: Evaluate the derivative of e^2x+ x.sinx

Solution:

y = e^2x + x.sinx

y’ = d(e^2x)/dx + d(x.sinx)/dx

To solve this we need to apply Chain rule for e^2x and Product rule for x.sinx

y’ = (d(e^2x)/dx . d(2x)/dx ) + (dx/dy. sinx + x . d(sinx)/dy)

y’ = e^2x.2 + (1.sinx + x.cosx)

y’ = 2.e^2x + sinx + xcosx

Practice Questions on Derivative of e^x

1. Find the derivative of e^5x

2. Find the derivative of x³.e^3x

3. Evaluate: (d/dx) [e^x/(x² + 2)]

4. Evaluate the derivative of: e^x. log x

Summary

The differentiation of the exponential function e^x is straightforward and unique in calculus, as its derivative is itself. Mathematically, this is expressed as [Tex]\frac{d}{dx} e^x = e^x[/Tex]. This means that the rate of change of e^x with respect to x is , e^x a property that makes the exponential function uniquely important in various fields of mathematics and applied sciences.

FAQs on Derivative of e^x

What is Derivative in Math?

In mathematics, the derivative of a function tells about the change in output with a change in input. It measures the instantaneous rate of change of the function at a specific point.

What is Derivative of e raised to negative x?

Derivative of e raised to negative x or e^-x is (-e^-x).

What is Derivative of a^x?

Derivative of a^x is a^x.ln a.

What is Derivative of x.e^2x?

Derivative of x.e^2x is x.2e^2x + e^2x or e^2x( 2x + 1).