Derivative of e^x

Derivative of e^x is 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.

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

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

(e^x)' = e^x

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:

• Calculus in Maths
• Differential Calculus
• Derivatives Formulas
• Derivative of Exponential Function
• 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:

Given y = e^2x

Using Chain Rule
⇒ 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:

Given: y = e^x/x²

Using Quotient Rule i.e.,
[(d[u/v]/dx) = [u’v – uv’]/v²]

⇒ 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