Section 4-7-2

Chapter 4
Differential Calculus and Its Uses

4.7 The General Power Rule

4.7.2 The Power Rule for Rational Exponents

Next we show that the Power Rule holds for all positive rational exponents, i.e., exponents of the form `r=ptext[/]q`, where `p` and `q` are positive integers. We have just worked out the case with `p=7` and `q=3`. Now we show that the same argument works for any `p` and `q`.

We introduce a name for the `r`-th power function, say, `u=t^r`. Then `u` is also equal to `t^(p//q)`. If we raise both sides of this equation to the `q`-th power, we have

u^{q} = t^{p} .

On the right in this equation we have a function of `t` whose derivative we know from the Power Rule, since `p` is a positive integer. On the left, we have the same function of `t` (because of the equality) written in a different form, namely, as the `q`-th power of some other function `u`. We know exactly what function `u` is, but that's not important at the moment.

From the positive integer version of the Power Rule, we know that

\frac{d}{d u} u^{q} = q u^{q - 1} .

Thus, the Chain Rule — with `u^q` as a function of `u` and `u` as a function of `t` — tells us that

\frac{d}{d u} u^{q} = \frac{d}{d u} u^{q} \frac{d u}{d t} = q u^{q - 1} \frac{d u}{d t} .

This gives us the derivative of the left-hand side of `u^q=t^p`, which must equal the derivative of the right-hand side, so

q u^{q - 1} \frac{d u}{d t} = p t^{p - 1} .

Now `u` is the function whose derivative we wanted to know, so we solve this equation for `(du)/(dt)` and use algebra to simplify:

`(du)/(dt)`	`=(pt^(p-1))/(qu^(q-1))`
	`=p/q(t^(p-1))/(u^(q-1))`
	`=p/q(t^(p-1))/((t^(p//q))^(q-1))`
	`=p/q (t^(p-1))/(t^((q-1)p//q))`
	`=p/q(t^(p-1))/(t^(p-p//q))`
	`=p/qt^(p-1-p+p//q)`
	`=p/qt^(-1+p//q)`
	`=rt^(r-1)`.

Activity 1

In the first step of the preceding calculation, we solved for `dutext[/]dt` by dividing both sides of the previous equation by `qu^(q-1).` Give reasons for the rest of the steps in the argument.

Comment on Activity 1

Activity 2

Combine the previous result — the Power Rule for positive rational powers — with the Power Rule for the `-1` power to derive the Power Rule for negative rational powers. Notation is important for making sense of this task. Suppose you want to differentiate `t^r` where `r` is a negative rational number. Then `r=-s`, where `s` is positive (in fact, `s` is the absolute value of `r`), so we know the Power Rule for `t^s`. If we write `u=t^r=t^(-s)`, then we can take reciprocals on both sides and write `u^(-1)=t^s.`

Comment on Activity 2

Activity 2 completes our justification that the Power Rule

\frac{d}{d t} t^{r} = r t^{r - 1}

is correct for every power function with a rational exponent `r`.

Contents for Chapter 4

Chapter 4 Differential Calculus and Its Uses

4.7 The General Power Rule

Chapter 4
Differential Calculus and Its Uses