Calculus - Product Rule

Equation



The equation is read: The derivative of (f * g) is equal to: the (derivative of f times g) + (f times the derivate of g).

Proof

Suppose

and that f and g are each differentiable at the fixed number x. Then

Now the difference

is the area of the big rectangle minus the area of the small rectangle in the illustration.

That L-shaped region can be split into two rectangles, the sum of whose areas is readily seen to be:

(The illustration disagrees with some special cases, since f(w) need not actually be bigger than f(x) and g(w) need not actually be bigger than g(x). Nonetheless, the equality of (2) and (3) is easily checked by algebra.)

Therefore the expression in (1) is equal to

If all four of the limits in (5) below exist, then the expression in (4) is equal to

Now

because f(x) remains constant as w → x;

because g is differentiable at x;

because f is differentiable at x;

and now the "hard" one:

because g is continuous at x. How do we know g is continuous at x? Because another theorem says differentiable functions are continuous.

We conclude that the expression in (5) is equal to

Example

y = sin(x) * x

1. First, derive sin(x) and x separately. sin(x) represents f in the equation for the product rule and x represents g.

• You should get cos(x) and 1 ( f' and g' respectively)

2. Now we have:
f = sin(x)
g = x
f' = cos(x)
g' = 1
3. Now plug in the variables into the Product Rule equation.

• You should get: y' = cos(x)*x + sin(x)
  

Going Further

If you know the product rule you never have to use the quotient rule.
For example:

y = sin(x)/x

1. First, rewrite the equation into the Product rule form.

• y = sin(x) * (1/x)
• Tip: 1/x is equal to x^-1
  

2. Now derive sin(x) and 1/x separately. sin(x) represents f in the equation for the product rule and 1/x represents g.

• You should get cos(x) and -x^-2 ( f' and g' respectively)

3. Now we have:
f = sin(x)
g = x^-1
f' = cos(x)
g' = -x^-2

4. Now plug in the variables into the Product Rule equation.

• You should get: y' = cos(x)*(x^-1) + sin(x)*(-x^-2)
• When simplified you should get: y' = 0