The rate of change [slope] of a function at a point
is the limiting value of its average slope
over an interval including that point,
as the width of the interval shrinks to zero:
All the remaining Laws and Rules can be proven by
algebraic manipulation of this definition.
- Power Law:
valid for all powers p, whether positive, negative,
integer, rational, irrational, real, imaginary or complex.
- Product Law:
The derivative of the product of two functions is not
the product of their derivatives! Instead,
- Constant times a Function:
The Product Law gives
where a is a constant (, not a function of x).
This is often referred to as
"pulling the constant factor outside the derivative."
- Function of a Function:
Suppose y is a function of x and x is in turn a function of t.
(also known as the Chain Rule)
where k is any constant.
- Natural Logarithms:
Jess H. Brewer