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Home Differentiation Chain Rule Chain Rule - Definition and Proof
 
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Chain Rule - Definition and Proof

CHAIN RULE

Let f, G be two real functions and define the new function g by the rule

g(t) = G(f(t)).

At any value of t where the derivatives f'(t) and G'(f(t)) exist, g'(t) also exists and has the value

g'(t) = G'(f(t))f'(t).

PROOF

Let

x = f(t), y = g(r), y = G(x).

Take t as the independent variable, and let Δf ≠ 0 be infinitesimal. Form the corresponding increments Δx and Δy. By the Increment Theorem for x = f (t), Δx is infinitesimal. Using the Increment Theorem again but this time for y = G(x), we have

Δy = G'(x) Δx + ε Δx

or some infinitesimal ε. Dividing by Δt,

02_differentiation-337.gif

Then taking standard parts,

02_differentiation-338.gif

or

02_differentiation-339.gif
Home Differentiation Chain Rule Chain Rule - Definition and Proof

Last Update: 2010-11-25