Understand the Chain Rule Formula
The chain rule formula is a very simple but important formula. Basically, we have u=g(x) which is derivable at 'x' and y=f(u) which is derivable at the corresponding value of (u). The rule says that the composite function y=f(g(x)) is derivable at x and the following fact is true:
Practically, you have to keep in mind that differentiation of 'y' with respect to 'x' equals the differentiation of 'y' with respect to 'u' multiplied by differentiation of 'u' with respect to 'x'.
Have a look at the chain rule proof
In order to make the chain rule proof, we take y = f(u) and u = g(x) and we have to demonstrate that:
We consider Δu and Δy as the increments of u and y respectively corresponding to the increment Δ x of 'x'.
We reach the following relation:
The limit Δu must tend to 0.
is equal to differentiation of y with respect to u.
This fact implies that:
Because u is derivable at x, it is continuous at x and:
Taking the limit as Δ x → 0 in (i), we obtain:
And with that, the proof of chain rule is ready. The formula and the demonstration itself are very important for this mathematical chapter.
The chain rule we just learned is very utile because it turns a complicate derivative into a series of easier derivatives. Basically, instead of being necessary to calculate (f ∘ g)′(t) which can be very difficult, the rule gives us a method of computing our expression in terms of f′ and g′. You always have the chance to solve the exercise by directly applying the definition of the derivative to compute the derivate of a composite function, but this possibility is usually very hard.
In conclusion, the chain rule formula and proof are two important basic facts of math which should be treated with maximum attention.
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