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Chain Rule

The Chain Rule is useful when several variables depend on each other. A typical case is where z depends on x and y, while x and y depend on another variable t. We shall call t the independent variable, x and y the intermediate variables, and z the dependent variable. Figure 11.5.1 shows which variables depend on which.

11_partial_differentiation-291.gif

Figure 11.5.1

CHAIN RULE

if z is a smooth function of x and y, while x and y are differentiate functions of t, then

dz/dt exists

and

11_partial_differentiation-292.gif

Discussion If z = F(x, y) and x = g(t), y = h(t), then z as a function of f is

z = f(t) = F(g(r), h(f)).

We can give a more precise statement of the Chain Rule using functional notation:

If g(t) and h(t) are differentiate at t0, and F(x, y) is smooth at (x0, y0) where x0 = g(t0) and y0 = h(t0), then f'(f0) exists

and

f'(t0) = Fx(x0, y0)g'(t0) + Fy(x0, y0) h'(t0).

We shall give some examples and then prove the Chain Rule.

Example 1

Example 2

Example 3: Increase Rate of Prices

PROOF OF THE CHAIN RULE

We use the Increment Theorem. Let Δt be a nonzero infinitesimal, and let Δx, Δy, and Δz be the corresponding increments of x, y and z. Then Δx and Δy are infinitesimal, and

11_partial_differentiation-302.gif

where ε1 and ε2 are infinitesimal. Dividing by Δt,

11_partial_differentiation-303.gif

Taking standard parts, we see that

11_partial_differentiation-304.gif
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Last Update: 2010-11-25