= Derivative =

A '''derivative''' is an instantaneous rate of change with respect to an input variable. It is a ratio of [[Calculus/Differential|differentials]].

<<TableOfContents>>

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== Rules ==

The basic rules/identities are:

||'''Rule'''                    ||'''Formulation'''                                  ||'''Defined for...'''   ||
||constants                     ||{{attachment:const.svg}}                           || ||
||constant factors              ||{{attachment:constfact.svg}}                       || ||
||polynomials ('''power rule''')||{{attachment:polynomial.svg}}                      || ||
||exponentiation                ||{{attachment:e.svg}}                               || ||
||exponentiation (generalized)  ||{{attachment:exp.svg}}                             ||''a > 0''              ||
||logarithms                    ||{{attachment:ln.svg}}                              ||''x > 0''              ||
||logarithms (generalized)      ||{{attachment:log.svg}}                             ||''x > 0'' and ''a > 0''||

For [[Calculus/Trigonometry|trigonometric functions]]:

||'''Rule'''     ||'''Formulation'''        ||'''Defined for...'''||
||sine           ||{{attachment:sin.svg}}   || ||
||cosine         ||{{attachment:cos.svg}}   || ||
||tangent        ||{{attachment:tan.svg}}   || ||
||inverse sine   ||{{attachment:arcsin.svg}}||''-1 < x < 1''      ||
||inverse cosine ||{{attachment:arccos.svg}}||''-1 < x < 1''      ||
||inverse tangent||{{attachment:arctan.svg}}|| ||



=== Chain Rule ===

For composite functions like ''e^2x^'' or ''sin(2x)'', the '''chain rule''' must be applied. Let ''f'' be the entire function as-is (e.g., ''e^2x^''), ''h'' be the 'inner function (e.g., ''2x''), and ''g'' be the 'outer' function (e.g., ''e^h(x)^'').

{{attachment:chain.svg}}

An inconvenient function ''h(x)'' can be rewritten as ''e^ln(h(x))^'', which can be evaluated using this rule.



=== Product Rule ===

Consider a function like ''f(x) = g(x)h(x)''. Evaluate as:

{{attachment:prod1.svg}}

This '''product rule''' holds for vector multiplication. That is, given a vector ''f'' that was defined as a [[Calculus/VectorOperations#Dot_Product|dot product]] like ''f = g ⋅ h'', the derivative (i.e., [[Calculus/SpeedVelocityAndAcceleration|over time]]) of ''f'' can be calculated from known derivatives of the components. In this case, try:

{{attachment:prod2.svg}}

The same is true of an ''f'' defined as a [[Calculus/VectorOperations#Cross_Product|cross product]] like ''f = g × h''. Try:

{{attachment:prod3.svg}}



=== Quotient Rule ===

Consider a function like ''f(x) = g(x)/h(x)''. Evaluate as:

{{attachment:quot.svg}}



=== Properties ===

Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'':

{{attachment:sum.svg}}.

The follows from the [[Calculus/Differential|total differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'':

''f = gh''

''df = f,,g,,dg + f,,h,,dh''

''df/dx = f,,g,,(dg/dx) + f,,h,,(dh/dx)''

And clearly the partial derivatives ''f,,g,,'' and ''f,,h,,'' 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.



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