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| Given a multivariate composition like ''f(g,h)'' where ''g = g(x,y)'' and ''h = h(x,y)'', and using a notation that expresses [[Calculus/PartialDerivative|partial derivatives]] as ''a,,b,, = ∂a/∂b'', the [[Calculus/Differential|total differential]] demonstrates that: | Equivalently, consider: {{attachment:notation1.svg}} {{attachment:notation2.svg}} === Multivariate === A notation for [[Calculus/PartialDerivative|partial derivatives]] will be used such that ''a,,b,, = ∂a/∂b''. Given a multivariate composition like ''f(g,h)'' where ''g = g(x,y)'' and ''h = h(x,y)'', the [[Calculus/Differential|total differential]] demonstrates that: |
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| Clearly then ''f,,g,,g,,x,, + f,,h,,h,,x,, = f,,x,,'' and ''f,,g,,g,,y,, + f,,h,,h,,y,, = f,,y,,'', so: | Clearly then ''f,,g,,g,,x,, + f,,h,,h,,x,, = f,,x,,'' and ''f,,g,,g,,y,, + f,,h,,h,,y,, = f,,y,,''. Equivalently, consider: {{attachment:notation3.svg}} |
Chain Rule
The chain rule is an approach for differentiating a composition of functions.
Contents
Description
Given a function defined like f(x) = g(h(x)), the derivative of f is g'(h(x)) h'(x).
Sometimes this is notated as f = g ∘ h and f' = (g' ∘ h)h' .
Equivalently, consider:
Multivariate
A notation for partial derivatives will be used such that ab = ∂a/∂b.
Given a multivariate composition like f(g,h) where g = g(x,y) and h = h(x,y), the total differential demonstrates that:
df = fgdg + fhdh
df = fg(gxdx + gydy) + fh(hxdx + hydy)
df = fggxdx + fggydy + fhhxdx + fhhydy
df = (fggx + fhhx)dx + (fggy + fhhy)dy
Clearly then fggx + fhhx = fx and fggy + fhhy = fy. Equivalently, consider:
Usage
Point elasticity is defined as:
It can be simplified using the chain rule into:
Start with the final formulation and use u- and v-substitution:
Note in particular that the equation for v can be rewritten in terms of P:
Now the final formulation can be rewritten as:
Several of these terms are trivial to derive. The derivative of e to an exponent is itself, so the derivative of P = ev with respect to v is P. The derivative of a natural logarithm of some variable is 1 over that variable, so the derivative of u = ln Q with respect to Q is 1/Q. This leaves the equation as:
