### Product Rule

Derivative of f(x) g(x) = f '(x) g(x) + f(x) g'(x)

Example: The derivative of x^{2} sin(x) = 2x sin(x) + x^{2}
cos(x)

### Quotient Rule

Derivative of f(x) / g(x) = ( g(x) f'(x) - f(x) g'(x) ) / (g(x))^{2}

A ditty that helps you remember this rule is "low d high - high d low all over low squared"

Example: The derivative of (2x^{2}-3x-9) / (2x+3) = ((2x+3)(4x-3) -
(2x^{2}-3x-9)(2))/(2x+3)^{2}

### Exponent Rule

Derivative of f(x)^{n} = n f(x)^{n-1} f'(x)

Note that the exponent rule is a special case of the chain rule, which is
described in the next section.

Example 1: The derivative of sin^{3}(x) = 3 sin^{2}(x) cos(x)

Example 2: n doesn't need to be a positive integer. It can be a
fraction, or even negative. This can be used to find the derivative of 1/sqrt(x),
which is x^{-1/2}.

The derivative of x^{-1/2} = (-1/2) x^{-3/2}.

### Chain Rule

Derivative of f(g(x)) = f'(g(x)) g'(x)

Example: The derivative of tan(sin(x)) = sec^{2}(sin(x)) cos(x)

### Internet references

HMC Mathematics Online Tutorial: The
Quotient Rule

Paul's online math notes: The
Chain Rule

1728 Software Systems: Product
Rule, Quotient Rule, and Chain rule, with examples.

### Related pages in this website

Limits

Definition of Continuous

The webmaster and author of this Math Help site is
Graeme McRae.