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

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

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

\frac{d(a^x)}{dx} = \lim_{h \to 0} \frac{a^{x+h} - a^x}{h}

= \lim_{h \to 0} \frac{a^x \cdot a^h - a^x}{h}

= \lim_{h \to 0} \frac{a^x \left(a^h - 1\right)}{h}

= a^x \lim_{h \to 0} \frac{a^h - 1}{h}

= a^x ln a {Since, \lim_{h \to 0} \frac{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)

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 ⇒ \frac{1}{x \ln a}

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 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})