Properties of the Exponential Function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of \(e ,\) we must verify that the usual laws of exponents hold for the function \(e^{x} .\)

Theorem 6.19

Properties of the Exponential Function

If \(p\) and \(q\) are any real numbers and \(r\) is a rational number, then

\(e^{p} e^{q} = e^{p + q}\)
\(\frac{e^{p}}{e^{q}} = e^{p - q}\)
\(\left(\right. e^{p} \left.\right)^{r} = e^{p r}\)

Proof

Note that if \(p\) and \(q\) are rational, the properties hold. However, if \(p\) or \(q\) are irrational, we must apply the inverse function definition of \(e^{x}\) and verify the properties. Only the first property is verified here; the other two are left to you. We have

\[\text{ln} \left(\right. e^{p} e^{q} \left.\right) = \text{ln} \left(\right. e^{p} \left.\right) + \text{ln} \left(\right. e^{q} \left.\right) = p + q = \text{ln} \left(\right. e^{p + q} \left.\right) .\]

Since \(\text{ln} x\) is one-to-one, then

\[e^{p} e^{q} = e^{p + q} .\]

□

As with part iv. of the logarithm properties, we can extend property iii. to irrational values of \(r ,\) and we do so by the end of the section.

We also want to verify the differentiation formula for the function \(y = e^{x} .\) To do this, we need to use implicit differentiation. Let \(y = e^{x} .\) Then

\[\begin{aligned} \text{ln} y & = & x \\ \frac{d}{d x} \text{ln} y & = & \frac{d}{d x} x \\ \frac{1}{y} \frac{d y}{d x} & = & 1 \\ \frac{d y}{d x} & = & y . \end{aligned}\]

Thus, we see

\[\frac{d}{d x} e^{x} = e^{x}\]

as desired, which leads immediately to the integration formula

\[\int e^{x} d x = e^{x} + C .\]

We apply these formulas in the following examples.

Example 6.38

Using Properties of Exponential Functions

Evaluate the following derivatives:

\(\frac{d}{d t} e^{3 t} e^{t^{2}}\)
\(\frac{d}{d x} e^{3 x^{2}}\)

Solution

We apply the chain rule as necessary.

\(\frac{d}{d t} e^{3 t} e^{t^{2}} = \frac{d}{d t} e^{3 t + t^{2}} = e^{3 t + t^{2}} \left(\right. 3 + 2 t \left.\right)\)
\(\frac{d}{d x} e^{3 x^{2}} = e^{3 x^{2}} 6 x\)

Checkpoint 6.38

Evaluate the following derivatives:

\(\frac{d}{d x} \left(\right. \frac{e^{x^{2}}}{e^{5 x}} \left.\right)\)
\(\frac{d}{d t} \left(\right. e^{2 t} \left.\right)^{3}\)

Example 6.39

Using Properties of Exponential Functions

Evaluate the following integral: \(\int 2 x e^{− x^{2}} d x .\)

Solution

Using \(u\)-substitution, let \(u = − x^{2} .\) Then \(d u = −2 x d x ,\) and we have

\[\int 2 x e^{− x^{2}} d x = − \int e^{u} d u = − e^{u} + C = − e^{− x^{2}} + C .\]

Checkpoint 6.39

Evaluate the following integral: \(\int \frac{4}{e^{3 x}} d x .\)

This lesson is part of:

Applications of Integration

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