# Product Rule

 Our Introductory Calculus tutorial software program contains over 40 topic areas. One of them is the Product Rule, and this page summarizes the main ideas of this topic. This page is intended for review, and is not a substitute for the interactive, self-paced tutorials of the MathTutor introductory calculus software program.

General Explanation

In this lesson we will learn how to differentiate the product of two functions. First, let's derive the rule.

We'll begin with the definition of a derivative:

```dy        f(x+Δx) - f(x)   = lim                dx   Δx→0       Δx
```

Now, let f(x) = u(x)v(x), that is, the product of u(x) and v(x).
Then, f(x+Δx) = u(x+Δx)•v(x+Δx).

```dy        u(x+Δx)v(x+Δx) - u(x)v(x)   = lim                           dx   Δx→0           Δx
```

Separating terms gives us:

```dy        u(x+Δx)v(x+Δx) - u(x)v(x)   = lim                           dx   Δx→0        Δx           Δx
```

We can add -u(x+Δx)v(x) + v(x)u(x+Δx), which is really 0, to the limit above:

```dy        u(x+Δx)v(x+Δx) - u(x+Δx)v(x) + v(x)u(x+Δx) - u(x)v(x)   = lim                                                       dx   Δx→0               Δx                        Δx
```

Let's look at the second term. If we factor out v(x) we get:

```dy        v(x)[u(x+Δx)-u(x)]   = lim                    dx   Δx→0          Δx
```

which is equal to v(x) • du/dx. The same can be done to the first term
to get u(x) • dv/dx.

As a general rule:

```d(u(x)•v(x))        d(v(x))        d(u(x))             = u(x)•        + v(x)•            dx               dx             dx
```

Sample Problem

Find dy/dx for the following:

```y = (6x3+1) • 4x5
```

Solution

Let u(x) = 6x3+1 and v(x) = 4x5. Then:

```dy           d(4x5)       d(6x3+1)   = (6x3+1)•       + 4x5•         = (6x3+1)•(20x4) + 4x5•(18x2)dx             dx           dx

= 180x7 + 20x4 + 72x7 = 192x7 + 20x4
```