How do you take the derivative of an exponential expression?

Category: Derivatives »

Course: Calculus »

Relevant MM Lessons:

Overview: Exponential Derivatives

This question has its own dedicated lesson ». Make sure to check it out if you need a more thorough read, or want to become an expert on how exponential derivatives show up on tests.

The Basic Case - Base $e$

One of the characteristics that makes $e$ such a special number is that the derivative of an exponential function is itself, if the base is $e$. Symbolically:$$\frac{d}{dx} \,\, e^x = e^x$$You should know this fact cold. We'll certainly use it often enough. More likely, however, we'll be asked to compute derivatives that involve constants of some sort.

$kx$ Exponents

Via the Chain Rule, it is a quick exercise to prove that the derivative of the exponential function base $e$ with a variable exponent expression of the form $kx$ is itself, multiplied by the constant $k$.$$\frac{d}{dx} \,\, e^{kx} = ke^{kx}$$This is true whether $k$ is a positive or negative, whole number or not.$$\frac{d}{dx} \,\, e^{2x} = 2e^{2x}$$$$\frac{d}{dx} \,\, e^{-4x} = -4e^{-4x}$$$$\frac{d}{dx} \,\, e^{\pi x} = \pi e^{\pi x}$$In fact, one of the most common derivatives we'll take is $e^{-x}$ (the case where $k$ is $-1$).$$\frac{d}{dx} \,\, e^{-x} = -e^{-x}$$

Other Bases

If the exponential function has a base number other than $e$, its derivative will behave similar to the base $e$ cases, but will always be multiplied by the natural log of the base.$$\frac{d}{dx} \,\, a^{x} = a^{x} \cdot \ln(a)$$If other constants are present, those rules still apply.$$\frac{d}{dx} \,\, 3 \cdot 5^x = 3\ln(5) 5^x$$$$\frac{d}{dx} \,\, 7^{-2x} = -2\ln(7) 7^{-2x}$$
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