Note 3.4 - Derivatives of Exponential and Logarithmic Functions

Note 3.4 - Derivatives of Exponential and Logarithmic Functions

1 Introduction

Let’s fill in the last piece of puzzle: the derivatives ofe^xand lnx. The results will strongly reflect the facts that exponential functions are the standard functions to describe quantities whose rate of change depend on the quantities themselves.

2 The Derivative of e

To differentiatee^x, we first derive the following limit:

h→0lim e^h−1

h = 1.

Then, the derivative of e^x follows easily from the power rules.

We see that the rate of change ofe^xis itself!

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3 The Derivative of ln x

The derivative of lnxcan be derived easily with our available tools:

4 Arbitrary Bases

With our definitions ofa^x and log_axin Note 1, their derivatives are now easy to derive.

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5 Examples

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