Shown below is a graph of the function [math]f\left(x\right)=sin^{-1}\left(x\right)[/math]. [br][br]Interact with the applet below for a minute by dragging the white point along the arcsine curve. [br][color=#666666][b]The graph of the derivative of this function will be traced out in gray. [/b][/color] [br][br]If you'd like, you can take a guess as to what the derivative function might be.[br][color=#1e84cc](Just don't look it up on another tab.) [/color][br][br]Hit the [color=#38761d][b]"Clear & Retry"[/b][/color] button to take another guess (if you'd like.) [br][br]Complete the activity questions that appear below the applet. [br]
For the function [math]y=sin^{-1}\left(x\right)[/math], notice how [i]y[/i] is written in terms of [i]x[/i]. [br]Rewrite this equation so that [i]x[/i] is written in terms of [i]y[/i].
For the equation you wrote in (1) above, use implicit differentiation to differentiate both sides of your equation with respect to [i]x[/i].
[math]\frac{dy}{dx}=\frac{1}{cos\left(y\right)}[/math]
Sketch a right triangle that has an acute angle with a measure of "[i]y". [br][/i]Label the 3 sides of this right triangle (in terms of [i]x[/i]) so that the equation you wrote for (1) is true.
Rewrite the expression you obtained in (2) for [math]\frac{dy}{dx}[/math] solely in terms of [i]x[/i]. [br]After doing so, enter THIS FUNCTION into the applet above to see how its graph compares with the gray trace. [br][br]Is it correct?
[b][color=#cc0000]Once you're finished please proceed to the next investigation: [br][/color][color=#0000ff][url=https://www.geogebra.org/m/CyfGg78P]Derivative of the Inverse Cosine Function.[/url][/color][/b]