Product Rule

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

Example: The derivative of x² sin(x) = 2x sin(x) + x² 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 (2x²-3x-9) / (2x+3) = ((2x+3)(4x-3) - (2x²-3x-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 sin³(x) = 3 sin²(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)) = sec²(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.