= 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                 ||{{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 ''f(x) = e^h(x)^'' or ''f(x) = sin(h(x))'', the '''chain rule''' must be applied. Consider the outer function (exponentiation and sine in these examples) to be a function ''g(x)''. It follows that:

''f'(x) = g'(h(x)) * h'(x)''



=== Product Rule ===

Consider a function like ''f(x) = g(x)h(x)''. The derivative is evaluated as:

{{attachment:prod.svg}}

This '''product rule''' holds for vector multiplication; that is, for a [[Calculus/VectorOperations#Dot_Product|dot product]]:

''f = g ⋅ h''

''df/dx = h ⋅ (dg/dx) + g ⋅ (dh/dx)''

...and also for a [[Calculus/VectorOperations#Cross_Product|cross product]]:

''f = g × h''

''df/dx = h × (dg/dx) + g × (dh/dx)''



=== Quotient Rule ===

Consider a function like ''f(x) = g(x)/h(x)''. The derivative is evaluated 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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