Derivative of Exponential Functions

It describes the rate of change in an exponential function with respect to the independent variable. The derivative of an exponential function like a^x (a > 0) is the same function multiplied by a constant (the natural log of a). For example, a^x becomes a^x ln a when differentiated.

It can be obtained through the first principles of differentiation, utilizing limit formulas, and its graph increases when a > 1 and decreases when 0 < a < 1.

The formula for the differentiation or derivative of the exponential function is:

• If f(x) = a^x then f'(x) = a^x ln a
• If f(x) = eˣ, then f'(x) = eˣ

Derivative of Exponential Function Proof

It can be proved by using the first principle, where we find the derivative of a function using the definition of limits. From the definition of limits, we know that

For the function f(x) = a^x, using the First Principle,

=

=

= a^x ln a {Since, }

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)

Taking a = e in a^x , our function will be f(x) = e^x

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

e^x ln(e) = e^x log_ee (ln is a natural logarithm with base e)

e^x log_ee = e^x (from logarithm formulas, log_aa = 1 )

Hence, f'(x) = e^x

Exponential Function Derivative Graph

The nature of the graph of exponential function changes when the value of 'a' increases or decreases with respect to 1.

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

Since 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.

Derivative of Exponential and Logarithmic Functions

Exponential Function and Logarithmic Function are inverse of each other with change in base.

The derivative of exponential and logarithmic functions is mentioned below:

• a^x ⇒ a^x ln a
• e^x ⇒ e^x
• ln x ⇒ 1/x
• log_ax ⇒

Solved 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 = e^x.log x

Here the function is in u.v form

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

dy/dx = e^x(log x)' + log x(e^x)'

⇒ dy/dx = e^x.1/x + log x.e^x

Practice Problems

1. Find the derivative of the function

2. Determine the derivative of the function

3. Calculate the derivative of the function

4. Find the derivative of the function

5. Determine the derivative of the function