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: chain-rule-formula

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:chain rule proof

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 chain rule proof

The limit Δu must tend to 0.
look at the chain rule prooflimits
is equal to differentiation of y with respect to u.

This fact implies that:chain rule formula examples

Because u is derivable at x, it is continuous at x and:how to use chain rule

Taking the limit as Δ x → 0 in (i), we obtain:proof of chain rule

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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