The derivative of the function is

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

The derivative of the function g(x)=arcsin⁡(x)g(x)=\arcsin(x) is g′(x)=11−x2g'(x)=\dfrac{1}{\sqrt{1-x^{2}}}. A student claims that g′ ⁣(35)=54.g'\!\left(\dfrac{3}{5}\right)=\dfrac{5}{4}. Respond to each part below. (a) Calculate g′ ⁣(35)g'\!\left(\dfrac{3}{5}\right), showing every algebraic step of your work without skipping intermediate simplifications. (b) Provide a geometric justification for the value of 1−(35)2\sqrt{1-\left(\dfrac{3}{5}\right)^{2}} obtained in part (a) by constructing or describing a right triangle whose sine is 35\dfrac{3}{5}. Clearly indicate which side length corresponds to the square-root expression. (c) Explain why the same geometric reasoning implies that g′(x)g'(x) is positive for every xx in the interval (−1,1)(-1,1).

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