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Wize AP Calculus (AB) Textbook > Differentiation 1: Basics & Fundamental Properties

Derivatives of Exponential Functions

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Derivatives of Exponential Functions

The Derivative of The Exponential Function

The derivative of the exponential function f(x)=exf(x)=e^{x} is:
(ex)′=ex\boxed{\quad (e^x)'=e^{x}\quad }


Wize Tip
It is the only function whose derivative is itself!

The Derivative of General Exponential Functions

For a given b>0b>0, the derivative of the exponential function bxb^xis:
(bx)′=bxln⁡b\boxed{\quad (b^{x})'=b^{x}\ln b\quad }

Note: Notice if we let b=eb=e we get the previous formula since ln⁡e=1\ln e = 1.

Wize Concept
The Product, Quotient, and Chain Rules still apply for Exponential Derivatives!

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Example: Exponential Derivatives


Given y=3exx\displaystyle y=\frac{3e^x}{\sqrt{x}} find y′(x)y'(x).


y′(x)=3exx−12x−12(3ex)x=3exx−3ex2xx\displaystyle y'(x)=\frac{3e^x\sqrt{x}-\frac{1}{2}x^{-\frac{1}{2}}\left(3e^x\right)}{x} \\ \text{} \\=\frac{3e^x\sqrt{x}-\frac{3e^x}{2\sqrt{x}}}{x}





=6xex−3ex2x32\displaystyle =\frac{6xe^x-3e^x}{2x^{\frac{3}{2}}}