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On which derivative rule is the substitution rule based?

Question:

On which derivative rule is the substitution rule based?

Substitution Rule:

Some expressions are not integrated using basic integrating techniques. While other techniques may be used to anti-differentiate these expressions, the substitution rule provides one good way to obtain their integral. The substitution rule is a technique of integration that uses substitution to express an integral with a particular variable into another in terms of the substituted variable. A formula for the substitution rule is given by

$$\int f(g(x)) g'(x) = \int f(u) du $$

where {eq}u = f(g(x) {/eq}

Answer and Explanation:

The substitution rule is a technique that can be used to obtain the indefinite integral. For a composite function {eq}f(g(x)) {/eq}, substituting {eq}u = g(x) {/eq} leads into an integral in terms of the substituted variable {eq}u {/eq}. Or,

{eq}\displaystyle \int f(g(x)) g'(x) = \int f(u) du, {/eq}

where {eq}u = g(x) {/eq}. This method of taking the integral of the composite function is very similar to the concept of the chain-rule used in differentiation, which is represented by

{eq}\displaystyle f'(g(x)) = f'(x) g'(x) {/eq}

As observed, both rules use the concept of composition of functions to express a complicated integral/derivative into a simpler one. Therefore, the derivative rule in which the substitution rule is based on, is the {eq}\boxed{\rm chain\ rule} {/eq}.


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