Question

For this question, I'm having trouble taking the inverse. Does the derivative need to be taken before the inverse is?

Answer 1

It's asking on the slope of it for the f⁻¹(x), so you have to get the inverse firstly, then we can simply get the derivative by implicit differentiation,

$\bf f(x)=4x^5-\cfrac{1}{x^4}\qquad \stackrel{inverse}{x=4y^5-\cfrac{1}{y^4}}\implies x=4y^5-y^{-4} \\\\\\ \stackrel{\textit{implicit differentiation}}{1=4\left(5y^4\frac{dy}{dx} \right)-\left(-4y^{-5}\frac{dy}{dx} \right)}\implies 1=20y^4\cfrac{dy}{dx}+4y^{-5}\cfrac{dy}{dx} \\\\\\ 1=\cfrac{dy}{dx}(20y^4+4y^{-5})\implies \cfrac{1}{20y^4+4y^{-5}}=\cfrac{dy}{dx} \\\\\\ \left. \cfrac{1}{20y^4+\frac{4}{y^5}}=\cfrac{dy}{dx} \right|_{3,1}\implies \cfrac{1}{20(1)^4+\frac{4}{(1)^5}}\implies \cfrac{1}{24}$