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Derivative of Exponential Functions

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Derivative of Exponential Function stands for differentiating functions which are expressed in the form of exponents. We know that exponential functions exist in two forms, ax where a is a real number r and is greater than 0 and the other form is ex where e is Euler’s Number and the value of e is 2.718 . . . On differentiating ax we will get ax ln a and on differentiating ex we will get ex.

In this article, we will learn about the derivative of the exponential function, the derivative of the exponential function formula, the derivative of the exponential function proof, and examples in detail. But before learning about the differentiation of exponential function we must know about exponential function.

Exponential Function Definition

Exponential Function is a function whose base is a constant and the exponent is a variable. For Example, ax where a is a real number, ex where e is a constant and its value is 2.718… It should be noted that if a function has a variable in the base and a constant in the exponent then it is not an Exponential Function. For Example, xa, 2xa, etc are not exponential functions for real numbers a and b.

What is Exponential Function Derivative?

Exponential Function Derivative also called Differentiation of Exponential Function refers to finding the rate of change in an exponential function with respect to the independent variable. The derivative of the exponential function ax where a > 0 is ax ln a and the derivative of the exponential function given as ex is ex where e = 2.718… In other words, we can say that if we have y = f(x) = ax then dy/dx = d{f(x)}/dx = ax ln a, and if we have y = f(x) = ex then dy/dx = d{f(x)}/dx = ex

Derivative of Exponential Function Formula

The formula for the differentiation or derivative of the exponential function is given as follows

If f(x) = ax then f'(x) = d{f(x)}/dx = d(ax)/dx = ax ln a

If f(x) = ex then f'(x) = d{f(x)}/dx = d(ex)/dx = ex

Learn more Differentiation Formula.

Proof of Derivative of Exponential Function

The derivative of the exponential function can be proved by using the first principle. In the First Principle Differentiation, we find the derivative of a function using the definition of limits. From the definition of limits, we know that

f'(x) = limh→0 f(x + h) – f(x) / h

Now we have function f(x) = ax

Now finding the derivative of f(x) = ax using the First Principle we have,

f'(x) = d(ax)/dx = limh→0 ax+h – ax / h

⇒ d(ax)/dx = limh→0 ax ⨯ ah – ax / h {From law of exponents we have, ax+h = ax ⨯ ah }

⇒ d(ax)/dx = limh→0 ax (ah – 1) / h

⇒ d(ax)/dx = ax limh→0 (ah – 1) / h

⇒ d(ax)/dx = ax ln a {Since, limh→0 (ah – 1) / h = ln a}

Hence, the derivative of the Exponential Function ax is the product of ax and the natural logarithm of a.

Derivative of e to the Power x (ex)

We know that the derivative of exponential function ax is given as ax ln a where a is a real number and greater than 0.

Now if we assume, a = e in ax as e is a real number and its value is 2.718… i.e. greater than zero then our function will be f(x) = ex

Now we have,

f(x) = ex

⇒ f'(x) = ex lne

We know that ln is a natural logarithm with base e i.e. logee

Now from logarithm formulas, we know that logaa = 1

Hence, ln e = logee = 1

Putting this value in f'(x) we get

f'(x) = ex ln e = ex

Derivative of e2x

We have f(x) = e2x which is an exponential function as it is in the form of ex but it is also a composite function consisting of two functions e2x and 2x. Hence, we will find the derivative of e2x using the chain rule.

Using the chain rule we know that the derivative of a composite function is given in the form of the derivative of the first function multiplied by the derivative of the second function.

f(x) = e2x ⇒ f'(x) = 2.e2x

Hence, the derivative of e2x is 2.e2x where e2x is derivative of e2x and 2 is derivative of 2x.

Exponential Function Derivative Graph

We know that an exponential function is defined as ax where a is a real number greater than zero. The nature of the graph of exponential function changes when the value of ‘a’ increases or decreases with respect to 1.
The nature of the graph of the derivative of exponential function is discussed below:

  • Graph is increasing if a > 1
  • Graph is decreasing if 0 < a < 1

Since we know that exponential function ax is only defined when a > 0 and derivative of a function is tangent to the graph or curve of that function. Hence, the graph of exponential function derivative is increasing for a > 0. The graph of the exponential function and its derivative for a > 1 and a < 1 is attached below:

Graph of derivative of exponential function

Derivative of Exponential and Logarithmic Functions

Exponential Function and Logarithmic Function are inverse of each other with change in base. The logarithmic forms of the exponential function are tabulated below

Exponential Form

Logarithmic Form

x = ay

y = logax

x = ey

y = logex or ln x

The derivative of exponential and logarithmic functions is mentioned below:

  • If f(x) = ax ⇒ f'(x) = ax ln a
  • If f(x) = ex ⇒ f'(x) = ex
  • If f(x) = ln x ⇒ f'(x) = 1/x
  • If f(x) = logax ⇒ f'(x) = 1/(x ln a)

Read More,

Derivative of Exponential Function Examples

Example 1: Find the derivative of esin x.

Solution:

Given that y = esin x

using chain rule

dy/dx = esin x(cos x)

Example 2: Differentiate elog x.

Solution:

Let y = elog x

then dy/dx = elog x.1/x

Now we know that from the property of logarithm that elog x = x

Hence, dy/dx = elog x.1/x = x.1/x = 1

Example 3: Find the derivative of ex.log x.

Solution:

y = ex.log x

here the function is in u.v form

Applying the product rule of differentiation we u.v’ + v.u’

dy/dx = ex(log x)’ + log x(ex)’

⇒ dy/dx = ex.1/x + log x.ex

FAQs on Derivative of Exponential Function

1. What is Exponential Function?

Exponential Function is a function in which a constant is raised to a variable power.

2. What is the Formula of Derivative of Exponential Function?

The formula for the derivative of exponential function f(x) = ax is ax.ln a and the derivative of ex is ex

3. What is the Derivative of Exponential x?

The derivative of exponential x is ex

4. What is the Derivative of e2x?

The derivative of e2x is 2.e2x

5. What is the Differentiation of Exponential Function?

The differentiation of exponential function ax is ax.ln a and the differentiation of ex is ex

6. What is the Derivative of e-x?

The derivative of e-x is -e-x.

7. What is the Derivative of 4x?

The derivative of 4x is 4x ln 4. [As derivative of ax = ax ln a]

8. What is the Derivative of 6x?

The derivative of 6x is 6x ln 6. [As derivative of ax = ax ln a]



Last Updated : 08 Sep, 2023
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