Section 4-4-2

Chapter 4
Differential Calculus and Its Uses

4.4 The Product Rule

4.4.2 A General Rule for Differentiating Products

As we noted in the Comment on Activity 1, the general product rule for differentiation has the following form.

The Product Rule (Functional Notation) If

`ftext[(]x text[)]=gtext[(]x text[)]htext[(]x text[)]`,

then

f^{'} (x) = g (x) h^{'} (x) + g^{'} (x) h (x) .

We may also describe this rule in terms of variables.

The Product Rule (Variable Notation) If

`u=gtext[(]x text[)]`, `v=htext[(]x text[)]`, and `w=gtext[(]x text[)]htext[(]x text[)]`,

then

\frac{d w}{d x} = u \frac{d v}{d x} + v \frac{d u}{d x} .

Let's see why this general rule holds. Suppose `ftext[(]x text[)]=gtext[(]x text[)]htext[(]x text[),]` where `g` and `h` are any functions that have derivatives. We attempt to find `f'text[(]x text[)]` — a rate of growth — as approximately a rise divided by a run. From a starting `x,` a run of `Delta x` changes the independent variable to `x+ Delta x.` This also changes the value of `f` from `ftext[(]x text[)]=gtext[(]x text[)]htext[(]x text[)]` to `ftext[(]x+Delta x text[),]` which is equal to `gtext[(]x+Delta x text[)]htext[(]x+Delta xtext[).]`

To understand how changes in `g` and `h` affect the product `f` we approximate those changes by local linearity:

Rise \approx slope \times run

g (x + Δ x) - g (x) \approx g^{'} (x) Δ x

and

h (x + Δ x) - h (x) \approx h^{'} (x) Δ x .

We solve these approximate equalities for the new values of `g` and `h`:

g (x + Δ x) \approx g (x) + g^{'} (x) Δ x

and

h (x + Δ x) \approx h (x) + h^{'} (x) Δ x .

Now we multiply the last two expressions to get an approximate description of the new value of `f`:

$f (x + Δ x)$	$= g (x + Δ x) h (x + Δ x)$
	$\approx [g (x) + g^{'} (x) Δ x] [h (x) + h^{'} (x) Δ x]$
	$= g (x) h (x) + g (x) h^{'} (x) Δ x + h (x) g^{'} (x) Δ x + g^{'} (x) h^{'} (x) {(Δ x)}^{2}$
	$= f (x) + g (x) h^{'} (x) Δ x + h (x) g^{'} (x) Δ x + g^{'} (x) h^{'} (x) {(Δ x)}^{2} .$

Next we approximate the rise in `f` by subtracting `ftext[(]x text[)]` from both sides:

f (x + Δ x) - f (x) \approx g (x) h^{'} (x) Δ x + h (x) g^{'} (x) Δ x + g^{'} (x) h^{'} (x) {(Δ x)}^{2} .

From the rise, we can calculate the rate of growth by dividing both sides by the run:

\frac{f (x + Δ x) - f (x)}{Δ x} \approx g (x) h^{'} (x) + h (x) g^{'} (x) + g^{'} (x) h^{'} (x) Δ x .

Finally, we consider what happens as the run, `Delta x`, shrinks to zero. On the left, we get `f'text[(]x text[)]`. On the right, nothing happens to the first two terms, but the third term approaches zero. And the approximation becomes exact:

f^{'} (x) = g (x) h^{'} (x) + h (x) g^{'} (x) .

In words, the Product Rule says: The derivative of a product is the first factor times the derivative of the second plus the second factor times the derivative of the first.

Contents for Chapter 4

Chapter 4 Differential Calculus and Its Uses

4.4 The Product Rule

Chapter 4
Differential Calculus and Its Uses