The most important exponential function is \(e^x\).

A special property of \(e^x\) is that its derivative is itself:

If the exponent is not just \(x\), then we use the chain rule.

So the derivative of the exponent is multiplied by the exponential function.

For a general exponential function \(a^x\), where \(a>0\) and \(a\ne1\):

This is useful for functions such as \(2^x\), \(3^x\), or \(10^x\).

If the exponent is a function of \(x\), use both the exponential rule and the chain rule:

Differentiate

\[ y=e^x \]

Since the derivative of \(e^x\) is itself: \[ \frac{dy}{dx}=e^x \]

Differentiate

\[ y=e^{3x} \]

Use the chain rule. The derivative of \(3x\) is \(3\).

Differentiate

\[ y=e^{x^2+1} \]

The derivative of the exponent \(x^2+1\) is \(2x\).

Differentiate

\[ y=2^x \]

Use \[ \frac{d}{dx}(a^x)=a^x\ln a \]

Differentiate

\[ y=5^{2x-1} \]

The derivative of the exponent \(2x-1\) is \(2\).

Sometimes exponential functions appear together with other functions.

Example:

\[ y=xe^x \]

Use the product rule:

\[ \frac{dy}{dx}=x(e^x)+e^x(1) \]

• Forgetting the chain rule for \(e^{f(x)}\).
• Writing \(\frac{d}{dx}(2^x)=x2^{x-1}\), which is incorrect.
• Forgetting the factor \(\ln a\) when differentiating \(a^x\).
• Not simplifying the final answer when factorising is possible.