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| 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: | 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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| ''f'(x) = g'(h(x)) * h'(x)'' | {{attachment:chain.svg}} |
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| Consider a function like ''f(x) = g(x)h(x)''. The derivative is evaluated as: | Consider a function like ''f(x) = g(x)h(x)''. Evaluate as: |
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| {{attachment:prod.svg}} | {{attachment:prod1.svg}} |
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| This '''product rule''' holds for vector multiplication; that is, for a [[Calculus/VectorOperations#Dot_Product|dot product]]: | 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: |
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| ''f = g ⋅ h'' | {{attachment:prod2.svg}} |
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| ''df/dx = h ⋅ (dg/dx) + g ⋅ (dh/dx)'' | The same is true of an ''f'' defined as a [[Calculus/VectorOperations#Cross_Product|cross product]] like ''f = g × h''. Try: |
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| ...and also for a [[Calculus/VectorOperations#Cross_Product|cross product]]: ''f = g × h'' ''df/dx = h × (dg/dx) + g × (dh/dx)'' |
{{attachment:prod3.svg}} |
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| Consider a function like ''f(x) = g(x)/h(x)''. The derivative is evaluated as: | Consider a function like ''f(x) = g(x)/h(x)''. Evaluate as: |
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 e2x or sin(2x), the chain rule must be applied. Let f be the entire function as-is (e.g., e2x), h be the 'inner function (e.g., 2x), and g be the 'outer' function (e.g., eh(x)).
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):
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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.