Derivative of the Exponential Function

Exponential functions are among the most important functions in mathematics. Their derivatives are straightforward but depend on the base. The most elegant case is the natural exponential function $e^x$.

📜 Formula for $e^x$

The natural exponential function has the simplest derivative of all:

$$ \frac{d}{dx} \big(e^x\big) = e^x $$

💡 Key property: $e^x$ is the only exponential function that is its own derivative.

📜 Formula for General Bases $a^x$

If the base is $a > 0, a \neq 1$, then the formula is:

$$ \frac{d}{dx} \big(a^x\big) = a^x \cdot \ln(a) $$

💡 Important: The additional factor $\ln(a)$ arises from the change of base. For $a = e$, this factor becomes $\ln(e) = 1$, so the formula simplifies to $\frac{d}{dx}(e^x) = e^x$.

🪜 Application in 3 Simple Steps

You can apply the exponential rule directly whenever you encounter $e^x$ or $a^x$. For more complex exponents, use the chain rule.

Identify the exponential form: Is it $e^x$ or a general base $a^x$?
Apply the formula: For $e^x$, the derivative is $e^x$. For $a^x$, the derivative is $a^x \ln(a)$.
Extend with the chain rule if necessary: If the exponent is not just $x$ but $u(x)$, then: $$\frac{d}{dx}\big(a^{u(x)}\big) = a^{u(x)}\ln(a)\cdot u'(x).$$

✏️ Examples for Illustration

Example 1: Natural Exponential Function

Given the function: $f(x) = e^x$

Derivative: $f'(x) = e^x$

Example 2: Exponential with Base 2

Given the function: $f(x) = 2^x$

Derivative: $f'(x) = 2^x \ln(2)$

Example 3: Exponential with Inner Function (Chain Rule)

Given the function: $f(x) = e^{3x}$

Derivative: $f'(x) = e^{3x}\cdot 3 = 3e^{3x}$

Example 4: General Base with Inner Function

Given the function: $f(x) = 5^{x^2}$

Derivative: $$ f'(x) = 5^{x^2} \ln(5) \cdot (2x) = 2x \cdot 5^{x^2}\ln(5). $$