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Size: 3066
Comment: Cleanup
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Comment: Generalized power rule
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| Deletions are marked like this. | Additions are marked like this. |
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| ||'''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''|| |
||'''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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| Consider a function like ''f(x) = g(x)h(x)''. Evaluated as: | Consider a function like ''f(x) = g(x)h(x)''. Evaluate as: |
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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 (power rule) |
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generalized power rule |
given f(x) = g(x)h(x), |
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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 |
![[ATTACH]](/Calculus/Derivative?action=AttachFile&do=upload_form&target=gen.svg&ticket=006a4cbbf3.1ba0d221a8f9a1e3f558c1f68694ae39abac1e8f "[ATTACH]"){kind=small}
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.