Derivative of Exponential Functions

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

Mathematically, the derivative of an exponential function is expressed as \frac{d(ax)}{dx} = (ax)' = ax \ln a . This derivative can be obtained through the first principles of differentiation, utilizing limit formulas. The graph of the derivative of an exponential function alters its direction when a > 1 and when a < 1.

In this article, we will learn about the derivative of the exponential function, its formula, proof of the formula, 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, a^x where a is a real number, e^x 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, x^a, 2x^a, etc are not exponential functions for real numbers a and b.

Read in Detail: Calculus in Maths

What is the Derivative of Exponential Function?

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

Derivative of Exponential Function Formula

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

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

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

Learn More: Differentiation Formula

Derivative of Exponential Function Proof

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) = lim_h→0 f(x + h) - f(x) / h

Now we have function f(x) = a^x

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

f'(x) = d(a^x)/dx = lim_h→0 a^x+h - ax / h

⇒ d(a^x)/dx = lim_h→0 a^x ⨯ a^h - a^x / h {From law of exponents we have, a^x+h = a^x ⨯ a^h }

⇒ d(a^x)/dx = lim_h→0 a^x (a^h - 1) / h

⇒ d(a^x)/dx = a^x lim_h→0 (a^h - 1) / h

⇒ d(a^x)/dx = a^x ln a {Since, lim_h→0 (a^{h - 1}) / h = ln a}

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

Derivative of e to the Power x (e^x)

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

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

Now we have,

f(x) = e^x

⇒ f'(x) = e^x lne

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

Now from logarithm formulas, we know that log_aa = 1

Hence, ln e = log_ee = 1

Putting this value in f'(x) we get

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

Derivative of e^2x

We have f(x) = e^2x which is an exponential function as it is in the form of e^x but it is also a composite function consisting of two functions e^2x and 2x. Hence, we will find the derivative of e^2x 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) = e^2x ⇒ f'(x) = 2.e^2x

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

Exponential Function Derivative Graph

We know that an exponential function is defined as a^x 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 a^x 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:

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

The derivative of exponential and logarithmic functions is mentioned below:

• If f(x) = a^x ⇒ f'(x) = a^x ln a
• If f(x) = e^x ⇒ f'(x) = e^x
• If f(x) = ln x ⇒ f'(x) = 1/x
• If f(x) = log_ax ⇒ f'(x) = 1/(x ln a)

People Also Read:

• Logarithmic Differentiation
• Implicit Differentiation
• Application of Derivative

Derivative of Exponential Function Examples

Example 1: Find the derivative of e^{sin x}.

Solution:

Given that y = e^{sin x}

using chain rule

dy/dx = e^{sin x}(cos x)

Example 2: Differentiate e^{log x}.

Solution:

Let y = e^{log x}

then dy/dx = e^{log x}.1/x

Now we know that from the property of logarithm that e^{log x} = x

Hence, dy/dx = e^{log x}.1/x = x.1/x = 1

Example 3: Find the derivative of e^x.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 = e^x.1/x + log x.e^x

Derivative of Exponential Functions Worksheet

1. Find the derivative of the function f'(x) = \frac{d}{dx}(e^x)

2. Determine the derivative of the function g'(x) = \frac{d}{dx}(3e^{2x})

3. Calculate the derivative of the function h'(x) = \frac{d}{dx}(e^{-2x})

4. Find the derivative of the function k'(x) = \frac{d}{dx}(4e^{-3x^2})

5. Determine the derivative of the function m'(x) = \frac{d}{dx}(2e^{3x + 1})

Conclusion

In conclusion, the derivative of the exponential function is a fundamental concept in calculus with wide-ranging applications in various fields such as physics, engineering, economics, and beyond. Understanding the rules for differentiating exponential functions, including those with different bases and those involving the chain rule, is crucial for tackling more complex mathematical problems.