Differentiation

Exponential Functions and their Derivatives (including Examples)

Exponential functions have the general form:

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with two constants a and b (called base). It’s quite common to use Euler’s number e = 2.7182… as the base and the exponential function expressed as such:

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with two constants a and c. Converting from one form to the other is not that difficult, just use ec = b or c = ln(b). Here’s how it works:

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As for the plot, you should keep two special cases in mind. For b > 1 (which corresponds to c > 0 in case of base e), the function goes through the point P(0,a) and goes to infinity as x goes to infinity.

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(Exponential function with b > 1 or c > 0. For example: f(x) = 8·3x)

This is exponential growth. When 0 < b < 1 (or c < 0) this turns into exponential decline. The function again goes through the point P(0,a), but approaches zero as x goes to infinity.

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(Exponential function with 0 < b < 1 or c < 0. For example: f(x) = 0.5x)

Here’s how the differentiation of exponential functions works. Given the function:

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 The first derivative is:

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For the case of base e:

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We get:

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You should remember both formulas. Note that the exponential functions have the unique property that their first derivative (slope) is proportional to the function value (height above x-axis). So the higher the curve, the sharper it rises. This is why exponential growth is so explosive.

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Example 1:

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Example 2:

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Example 3:

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Example 4:

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If exponential functions are combined with power or polynomial functions, just use the sum rule.

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Example 5:

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Example 6:

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(This was an excerpt from the FREE ebook “Math Shorts – Derivatives”)

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New Release for Kindle: Math Shorts – Derivatives

The rich and exciting field of calculus begins with the study of derivatives. This book is a practical introduction to derivatives, filled with down-to-earth explanations, detailed examples and lots of exercises (solutions included). It takes you from the basic functions all the way to advanced differentiation rules and proofs. Check out the sample for the table of contents and a taste of the action. From the author of “Mathematical Shenanigans”, “Great Formulas Explained” and the “Math Shorts” series. A supplement to this book is available under the title “Exercises to Math Shorts – Derivatives”. It contains an additional 28 exercises including detailed solutions.

Note: Except for the very basics of algebra, no prior knowledge is required to enjoy this book.

Table of Contents:

– Section 1: The Big Picture

– Section 2: Basic Functions and Rules

Power Functions
Sum Rule and Polynomial Functions
Exponential Functions
Logarithmic Functions
Trigonometric Functions

– Section 3: Advanced Differentiation Rules

I Know That I Know Nothing
Product Rule
Quotient Rule
Chain Rule

– Section 4: Limit Definition and Proofs

The Formula
Power Functions
Constant Factor Rule and Sum Rule
Product Rule

– Section 5: Appendix

Solutions to the Problems
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