If F(x) is an antiderivative of f(x) , then (1/a) F(ax) is an antiderivative of ...


If {eq}F(x) {/eq} is an antiderivative of {eq}f(x) {/eq}, then {eq}(1/a) F(ax) {/eq} is an antiderivative of {eq}f(ax) {/eq}.


Reversing the Chain Rule

When we find the derivative of a function, we apply a variety of differentiation rules, such as the Chain Rule, to yield the derivative. Finding the antiderivative is similar, except that we need to reverse these rules. Thus, we need to reverse the Chain Rule, which is applying substitution, to find an antiderivative.

Answer and Explanation:

If this statement is true, then differentiating the expression {eq}\frac{1}{a} F(ax) {/eq} should give the expression {eq}f(ax) {/eq}. Let's see if this is indeed the case. To do so, we need to apply the Chain Rule to this expression. As F is the antiderivative of f, that means f is the derivative of F.

{eq}\begin{align*} \frac{d}{dx} \frac{1}{a} F(ax) &=\frac{1}{a} f(ax) \cdot \frac{d}{dx} ax\\ &=\frac{1}{a} f(ax) \cdot a\\ &= f(ax) \end{align*} {/eq}

Thus, this is a true statement.

Learn more about this topic:

Antiderivative: Rules, Formula & Examples

from Calculus: Help and Review

Chapter 8 / Lesson 12

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