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| === 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}} |
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| 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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| 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)'': | {{attachment:sum.svg}}. The follows from the [[Calculus/Differential|total differential]]; substitute ''g'' and ''h'' for ''g(x)'' and ''h(x)'': |
Derivative
A derivative is an instantaneous rate of change with respect to an input variable. It is a ratio of differentials.
Rules
The basic rules/identities are:
Rule |
Formulation |
Defined for... |
constants |
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constant factors |
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polynomials |
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exponentiation |
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exponentiation (generalized) |
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a > 0 |
logarithms |
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x > 0 |
logarithms (generalized) |
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x > 0 and a > 0 |
Rule |
Formulation |
Defined for... |
sine |
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cosine |
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tangent |
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inverse sine |
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-1 < x < 1 |
inverse cosine |
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-1 < x < 1 |
inverse tangent |
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Chain Rule
For composite functions like f(x) = eh(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:
This product rule holds for vector multiplication; that is, for a dot product:
f = g ⋅ h
df/dx = h ⋅ (dg/dx) + g ⋅ (dh/dx)
...and also for a 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:
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 = 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.