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Chain Rule and 'The Mob'...Pretty darn good explanation

PhysicsRus

New member
Jan 1, 2013
1

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
his analogy is apt, but some of his math is wrong.

the formula:

$h'(c) = (g \circ f)'(c) = g'(f(c))\cdot f'(c)$

is correctly expressed as:

$$\left(\lim_{f(x) \to f(c)} \dfrac{g(f(x)) - g(f(c))}{f(x) - f(c)}\right)\left(\lim_{x \to c} \dfrac{f(x) - f(c)}{x - c}\right)$$

whereas in the video he writes:

$$h'(c) = \lim_{x \to c} \dfrac{g(f(x)) - g(f(c))}{x - c} \cdot \dfrac{f(x) - f(c)}{x - c}$$

that is he has the denominator wrong in the first factor, so we can't "cancel" the f(x) - f(c) term to obtain:

$$h'(c) = \lim_{x \to c} \dfrac{g(f(x)) - g(f(c))}{x - c} = \lim_{x \to c} \dfrac{h(x) - h(c)}{x - c}$$

an important point is also glossed over:

f has to be differentiable, and differentiable means continuous, so f(x)-->f(c) as x-->c. this is KEY.

i give him credit for his vivid analogy, but algebra counts, too.