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Home / Mathematics / Derivative of Exponential Function

SAT Derivative of Exponential Function Formulas with Solved Examples

Last Updated on Mar 21, 2025
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Derivative of Exponential Function

Suppose and represent a constant and a variable respectively then the exponential function is written as in mathematics. The derivative of raised to the power of with respect to is written in the following form in calculus:

The differentiation or derivative of the function with respect to is denoted as and is also simply written as .

The derivative of raised to the -th power with respect to is equal to the product of to the -th power and the natural logarithm of , i.e.

It is known as the differentiation rule of exponential function and it is used to find the derivative of any exponential function.

For example: Evaluate

Here the constant

Now, substitute it in the differentiation rule of exponential function to find its derivative.

Hence, it can be used as a formula to find the differentiation of any function in exponential form.

Important points:

  • One of the popular forms of the exponential function is , where ‘’ is “Euler’s number” and approximately. The derivative of exponential function is given by .
  • Exponential function is not defined for and so does the derivative of exponential function.

Derivative of Exponential Function Formula

The formula for derivative of exponential function is given by:

  • , then , or .
  • , then , or .

Partial Derivative of Exponential Function:

In general when we’re given a function , then the derivative of the function is given as

.

Now how’s that changed when you have a function of two variables, and this is where partial derivatives come into picture. Let’s understand partial derivatives of exponential function with the help of examples.

Example 1: .

First find the first order partial derivative with respect to (this means that in the above function is treated as just a number), is given as

.

Now find the first order partial derivative with respect to (this means that in the above function is treated as just a number), is given as

.

Example 2: .

.

.

Let’s look at second order partial derivatives for this example.

First find the second order partial derivative with respect to ,

Use the product rule to solve the above second order partial derivative, we get

 

Similarly, we can find the mixed second order partial derivative with respect to , i.e.

Derivative of Exponential Function Proof using First Principle:

Now we prove that the derivative of exponential function is using the first principle of differentiation.

First write the derivative of this function in limit form by the definition of the derivative,

We know that,

We know that,

Hence we have derived the derivative of exponential function using the first principle of derivatives.

Graph of Derivative of Exponential Function

The graph of the exponential function is increasing when whereas is decreasing when . Thus, the graph of exponential function

  • increases when
  • decreases when

Since the graph of derivative of exponential function is the slope function of the tangent to the graph of the exponential function, therefore the graph of derivative of exponential function is increasing for . Given below is the graph of the exponential function and the graph of the derivative of the exponential function for and .

Derivative of Exponential Function Solved Examples

1.Find the derivative of .

Solution: We know that

 

Here and

Therefore, the derivative of is or .

2.Find the derivative of .

Solution: We know that

Therefore, the derivative of is .

3.Find the derivative of .

Solution: We know that

Let , where

Substitute , we get

Therefore the derivative of is .

We hope this article has been helpful in enhancing your understanding and preparing you for U.S. competitive exams like the SAT, ACT, and AP Calculus. Stay connected with reliable resources for updates on related math topics and other essential subjects. Additionally, consider exploring practice tests and test series to assess your knowledge and improve your performance on exam day.

Derivative of Exponential Function FAQs

The derivative of exponential function , is the product of exponential function and natural log of , that is, .

The derivative of the exponential function is given by . We know that and if , the derivative of exponential function is given by .

Yes, exponential functions are differentiable everywhere.

Exponential functions are always continuous because they are always differentiable and continuity is a necessary (but not sufficient) condition for differentiability.

The exponential function is neither even nor odd. It is the sum of an even and odd function.
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