Question

Here is a "proof" of the chain rule:

(g∘f)′(x₀)= lim g(f(x))−g(f(x₀)/ x−x₀)
x→x₀

=lim g(f(x))−g(f(x₀)/(f(x)−f(x₀))⋅ f(x)−f(x₀)/ x−x₀)
x→x₀
=lim g(f(x))−g(f(x₀)/ f(x)−f(x₀))⋅lim f(x)−f(x₀)/ x−x₀)
x→x₀ x→x₀
=g′(f(x₀))f′(x₀).
What is wrong with this "proof"?

Answer 1

The mistake in the 'proof' of the chain rule lies in incorrectly separating the limits of the quotient rule, lim((g(f(x))-g(f(x₀)) / (f(x) - f(x₀))) * lim((f(x) - f(x₀)) / (x - x₀)). The correct application of the chain rule avoids this assumption.

Step-by-step explanation:

The mistake in this 'proof' of the chain rule lies in the incorrect application of the limit of the quotient rule. Let's analyze it step by step to see where the error occurs:

1. It starts correctly by writing the derivative of the composition as (g∘f)'(x₀).

2. Then, it uses the limit definition of the derivative to rewrite it as lim((g(f(x))-g(f(x₀)) / (x-x₀)) as x approaches x₀.

3. Here comes the mistake: it tries to rewrite the limit as lim((g(f(x))-g(f(x₀)) / (f(x) - f(x₀))) * lim((f(x) - f(x₀)) / (x - x₀)) as x approaches x₀. This step is incorrect because it is assuming that the limits of the numerator and the denominator can be separated, which is not generally true.

4. Finally, it wrongly concludes that the derivative of the composition is g'(f(x₀)) * f'(x₀).

Therefore, the mistake lies in the assumption made in step 3, where the limits were incorrectly separated. The correct application of the chain rule should not involve separating the limits in this manner.