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| {{attachment:const.svg}} | The basic rules/identities are: |
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| {{attachment:constfact.svg}} | ||'''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''|| |
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| {{attachment:polynomial.svg}} | For [[Calculus/Trigonometry|trigonometric functions]]: |
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| {{attachment:e.svg}} | ||'''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}}|| || |
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| {{attachment:exp.svg}}, for ''a > 0'' | |
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| {{attachment:ln.svg}}, for ''x > 0'' | |
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| {{attachment:log.svg}}, for ''x > 0'' and ''a > 0'' | === Properties === |
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| {{attachment:sin.svg}} | Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'', {{attachment:sum.svg}}. |
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| {{attachment:cos.svg}} | The '''product rule''' states that, given a function defined like ''f(x) = g(x)h(x)'', {{attachment:prod.svg}}. This follows from the [[Calculus/Differential|total differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'': |
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| {{attachment:tan.svg}} | ''f = gh'' |
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| {{attachment:arcsin.svg}}, for ''-1 < x < 1'' | ''df = f,,g,,dg + f,,h,,dh'' |
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| {{attachment:arccos.svg}}, for ''-1 < x < 1'' | ''df/dx = f,,g,,(dg/dx) + f,,h,,(dh/dx)'' |
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| {{attachment:arctan.svg}} | 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. |
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 |
|
|
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.