Exponential Derivative Calculator

Published on: July 17, 2026
Final Answer: Free • Full Steps: Plus

This Exponential Derivative Calculator differentiates functions where the variable sits in the exponent, and shows the working line by line. The natural exponential is its own derivative, and any other base picks up a factor of the natural logarithm of that base. That extra factor is what separates the two cases. Following the same pattern every time keeps the base factor from being forgotten.

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

\(\begin{gathered} \frac{d}{dx}\left(e^{x}\right) = e^{x} \\ \frac{d}{dx}\left(a^{x}\right) = a^{x} \ln\left(a\right) \end{gathered}\)
See Example 1 Hide Example 1

Problem

\(f\left(x\right) = e^{x}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the exponential function and any constant multiple.

In this problem: The function is \(e^{x}\).

\(f\left(x\right) = e^{x}\)

Step 2

Step 2 - Write the derivative rule for that exponential function.

In this problem: The derivative of \(e^{x}\) is itself.

\(\frac{d}{dx}\left(e^{x}\right) = e^{x}\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the exponential with its derivative and simplify.

\(f'\left(x\right) = e^{x}\)

Final Answer

\(f'\left(x\right) = e^{x}\)
See Example 2 Hide Example 2

Problem

\(f\left(x\right) = 3^{x}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the derivative rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front.

Step 1

Step 1 - Identify the exponential function and any constant multiple.

In this problem: The function is \(3^{x}\).

\(f\left(x\right) = 3^{x}\)

Step 2

Step 2 - Write the derivative rule for that exponential function.

In this problem: The derivative of \(3^{x}\) is the function times the natural logarithm of the base \(3\).

\(\frac{d}{dx}\left(a^{x}\right) = a^{x} \ln\left(a\right)\)

Step 3

Step 3 - Apply the rule, keeping the constant multiple in front.

In this problem: Replace the exponential with its derivative and simplify.

\(f'\left(x\right) = 3^{x} \ln\left(3\right)\)

Final Answer

\(f'\left(x\right) = 3^{x} \ln\left(3\right)\)

Example Problems

Click one to load it into the box and solve it.