proof of chain rule

Let’s say that g is differentiableMathworldPlanetmathPlanetmath in x0 and f is differentiable in y0=g(x0). We define:

φ(y)={f(y)-f(y0)y-y0if yy0f(y0)if y=y0

Since f is differentiable in y0, φ is continuousMathworldPlanetmath. We observe that, for xx0,


in fact, if g(x)g(x0), it follows at once from the definition of φ, while if g(x)=g(x0), both members of the equation are 0.

Since g is continuous in x0, and φ is continuous in y0,



(fg)(x0) = limxx0f(g(x))-f(g(x0))x-x0
= limxx0φ(g(x))g(x)-g(x0)x-x0
= f(g(x0))g(x0).
Title proof of chain rule
Canonical name ProofOfChainRule
Date of creation 2013-03-22 12:41:48
Last modified on 2013-03-22 12:41:48
Owner n3o (216)
Last modified by n3o (216)
Numerical id 6
Author n3o (216)
Entry type Proof
Classification msc 26A06