Exponential Derivative Calculator

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

This Exponential Derivative Calculator helps you differentiate exponential functions such as e^x, 2e^x, or 3^x. The natural exponential e^x is its own derivative, while a general base a^x picks up a factor of ln(a).

Step-by-step method

  1. Identify the exponential function and any constant multiple.
  2. Write the derivative rule for that exponential function.
  3. Apply the rule, keeping the constant multiple in front.

Formula:

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

Example 1:

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

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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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:

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