Page 23 - Calc for the Clueless

F. If y = csc x, y' = -cot x csc x.

Rule 7

The product rule. If y = f(x)g(x), then y' = f(x)g(x)' + g(x)f(x)'. The product rule says the first multiplied by the
derivative of the second added to the second multiplied by the derivative of the first.

Example 8—

Find y' if

Multiplying and combining like terms, y'= 15X + 34x + 11.
2

We could, of course multiply this example out.

2
Then y' = 15x + 34x + 11 as before.
However, later on the examples will be much longer or even impossible to multiply out. It is a blessing that we
have the product rule and the next two rules.

Rule 8

The quotient rule.

If y = f(x)/g(x), then

The quotient rule says the bottom times the derivative of the top minus the top times the derivative of the
bottom, all divided by the bottom squared.

Example 9—

Find y' if .

When simplifying, do not multiply out the bottom. Only multiply and simplify the top. You may simplify the
top by factoring, as we will do in other problems.

Simplified, .

Rule 9