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Derivatives
This page will cover derivatives and their applications, as well as some advanced limits. The Fundamental Theorem of Calculus is covered on the integral page.
Definition
, where
is a function continuous in
, and
is an arbitrary constant such that
.
- Multiple derivatives are taken by evaluating the innermost first, and can be notated as follows:
.
- The derivative of
can also be expressed as
, or the
th derivative of
can be expressed as
.
Basic Facts
for a constant
.
The Power Rule
Rolle's Theorem
If is differentiable in the open interval
, continuous in the closed interval
, and if
, then there is a point
between
and
such that
Extension: Mean Value Theorem
If is differentiable in the open interval
and continuous in the closed interval
, then there is a point
between
and
such that
.
L'Hopital's Rule
Note that this inplies that for any
.
Taylor's Formula
Let be a point in the domain of the function
, and suppose that
(that is, the
th derivative of
) exists in the neighborhood of
(where
is a nonnegative integer). For each
in the neighborhood,
where is in between
and
.
Chain Rule
If , then
Applications
- The slope of
at any given point is the derivative of
. (The obvious one.)
- Acceleration is the derivative of velocity in relation to time; velocity is the derivative of position in relation to time.
- The derivative of work (in Joules) in relation to time is power (in watts).