Exponential functions have the general form:

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:

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

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.

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

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.

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

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

The first derivative is:

For the case of base e:

We get:

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:**

**Example 2:**

**Example 3:**

**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:**

**Example 6:**

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