Question

f(x) = √(9 - x)
f^-1(x) = -x^2 + 9 (change the variables, solve for y)
dy/dx f^-1(x) = -2x (power rule)

This question seemed a little too easy, so I want to make sure it's right! I took the inverse and then took the derivative. It would have been incorrect to take the derivative and then the inverse, right?

Answer 1

you're correct.

$\bf y=\sqrt{9-x}\implies \stackrel{inverse}{x=\sqrt{9-y}}\implies x^2=9-y\implies y=9-x^2 \\\\\\ \cfrac{dy}{dx}=0-2x^1\implies \cfrac{dy}{dx}=-2x$