Derivative of Exponential Functions

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

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.

➢Practice: https://write.geeksforgeeks.org/post/6084929