Derivative
A derivative is an instantaneous rate of change with respect to an input variable.
Contents
Rules
The basic rules/identities are:
Rule |
Formulation |
Defined for... |
constants |
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|
constant factors |
|
|
polynomials |
|
|
exponentiation |
|
|
exponentiation (generalized) |
|
a > 0 |
logarithms |
|
x > 0 |
logarithms (generalized) |
|
x > 0 and a > 0 |
Rule |
Formulation |
Defined for... |
sine |
|
|
cosine |
|
|
tangent |
|
|
inverse sine |
|
-1 < x < 1 |
inverse cosine |
|
-1 < x < 1 |
inverse tangent |
|
|
Properties
Derivatives are linear: given a function defined like f(x) = αg(x) + βh(x), 
.
The product rule states that, given a function defined like f(x) = g(x)h(x), 
. This follows from the total differential; substitute g and h for g(x) and h(x):
f = gh
df = fgdg + fhdh
df/dx = fg(dg/dx) + fh(dh/dx)
And clearly the partial derivatives fg and fh are equal to h and g respectively, giving:
df/dx = h(dg/dx) + g(dh/dx)
Substituting back in the original functions gives the product rule.