Many exponential function models do not have Euler's number, \(e\), as the base.

For example, in biological models, \(2\) is often used as the base for exponential functions because it provides an easy way to represent a population's “doubling time.”

Another example is the decay of radioactive substances, which is usually expressed by the half-life, leading mathematicians to use a base of \(\dfrac{1}{2}\).

Let \( y = a^x \) where \(a\gt 0\) is a constant and \(a\in\mathbb{R}\).

We begin by taking the \( \ln \) of both sides which yields

Since \(a\) is a positive constant, this equation could be written as \(\ln(y) = (\ln(a))x\).

Implicitly differentiating both sides with respect to \( x \) gives

Solving for \(\dfrac{dy}{dx}\) gives

Since \( y = a^x \),

Note:

The general form for the derivative of any exponential function will work if \( a = e \).

If \(f(x)=e^x\), then the general form will give \( f'(x) = e^{x}\ln(e)\) and since \( \ln(e) = 1 \), we have \( f'(x) = e^{x}(1)= e^{x}.\)

Differentiate \( y = 3^x \).

Solution