AP Calculus AB and BC glossary

Exponential function

An exponential function has the variable in the exponent and a constant base. Its defining property in calculus is that its rate of change is proportional to its current value, which is why e to the x is its own derivative.

ddxex=ex,ddxax=axln⁡a\frac{d}{dx}e^{x} = e^{x}, \quad \frac{d}{dx}a^{x} = a^{x}\ln a

That self-reproducing property is exactly what makes exponentials the solutions of growth and decay equations, where the derivative is a constant multiple of the function.

The mistake

Applying the power rule to a to the x. The power rule needs the variable in the base; with the variable in the exponent the derivative brings down a factor of ln a instead.

Appears in: Unit 2: Defining the Derivative, Unit 7: Differential Equations