Chain Rule

The chain rule is an approach for differentiating a composition of functions.

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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 a_b = ∂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 = f_gdg + f_hdh

df = f_g(g_xdx + g_ydy) + f_h(h_xdx + h_ydy)

df = f_gg_xdx + f_gg_ydy + f_hh_xdx + f_hh_ydy

df = (f_gg_x + f_hh_x)dx + (f_gg_y + f_hh_y)dy

Clearly then f_gg_x + f_hh_x = f_x and f_gg_y + f_hh_y = f_y. Equivalently, consider:

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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 = e^v 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:

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CategoryRicottone