Differences between revisions 3 and 12 (spanning 9 versions)

Derivative

A derivative is an instantaneous rate of change with respect to an input variable. It is a ratio of differentials.

Contents

Derivative
1. Rules

Rules

The basic rules/identities are:

Rule	Formulation	Defined for...
constants
constant factors
polynomials (power rule)
generalized power rule	given f(x) = g(x)^h(x),
exponentiation
exponentiation (generalized)		a > 0
logarithms		x > 0
logarithms (generalized)		x > 0 and a > 0

For trigonometric functions:

Rule	Formulation	Defined for...
sine
cosine
tangent
inverse sine		-1 < x < 1
inverse cosine		-1 < x < 1
inverse tangent

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)).

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:

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

The same is true of an f defined as a cross product like f = g × h. Try:

Quotient Rule

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

Properties

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

The follows from the total differential; substitute g and h for g(x) and h(x):

f = gh

df = f_gdg + f_hdh

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.

CategoryRicottone

-  ⇤ ← Revision 3 as of 2025-10-16 19:41:02 → 
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  Editor: DominicRicottone
  Comment: Properties
+   ← Revision 12 as of 2026-06-08 21:48:39 → ⇥
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  Editor: DominicRicottone
  Comment: Note
-Deletions are marked like this.
+Additions are marked like this.
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-A '''derivative''' is an instantaneous rate of change with respect to an input variable.
+A '''derivative''' is an instantaneous rate of change with respect to an input variable. It is a ratio of [[Calculus/Differential|differentials]].
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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 ('''power rule''')||{{attachment:polynomial.svg}}                      || ||
||'''generalized power rule'''  ||given ''f(x) = g(x)^h(x)^'', {{attachment:gen.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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+Line 25:
-{{attachment:polynomial.svg}}
+For [[Calculus/Trigonometry|trigonometric functions]]:
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+Line 27:
-{{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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+Line 35:
-{{attachment:exp.svg}}, for ''a > 0''
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+Line 36:
-{{attachment:ln.svg}}, for ''x > 0''
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+Line 37:
-{{attachment:log.svg}}, for ''x > 0'' and ''a > 0''
+=== Chain Rule ===
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+Line 39:
-See the trigonometric functions' defined [[Calculus/Trigonometry|here]].
+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)^'').
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+Line 41:
-{{attachment:sin.svg}}
+{{attachment:chain.svg}}
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+Line 43:
-{{attachment:cos.svg}}
+An inconvenient function ''h(x)'' can be rewritten as ''e^ln(h(x))^'', which can be evaluated using this rule.
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+Line 45:
-{{attachment:tan.svg}}
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+Line 46:
-{{attachment:arcsin.svg}}, for ''-1 < x < 1''
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+Line 47:
-{{attachment:arccos.svg}}, for ''-1 < x < 1''
+=== Product Rule ===
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+Line 49:
-{{attachment:arctan.svg}}
+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}}
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+Line 73:
-Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'', {{attachment:sum.svg}}.
+Derivatives are linear: given a function defined like ''f(x) = αg(x) + βh(x)'':
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+Line 75:
-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|differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'':
+{{attachment:sum.svg}}.

The follows from the [[Calculus/Differential|total differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'':
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+Line 91:
-Because the derivative of a constant is 0, the product rule also proves that ''(αf)' = αf' ''.

Diff for "Calculus/Derivative"