Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 4 - Integration - 4.1 An Overview Of The Area Problem - Exercises Set 4.1 - Page 270: 4

Answer

$$A^{\prime}(x) = x$$

Work Step by Step

The area of the triangle equals half the base multiplied by the height, and the height equals $x$ while the half base equals $x$, so the area will be $$A = \dfrac{1}{2} (x)(x) = \dfrac{x^2}{2}$$ If we differentiate the area, we will get the function of $x$ as requested: $$A^{\prime}(x)=\frac{d}{d x} \left(\dfrac{x^{2}}{ 2}\right) =x=f(x)$$
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