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(dy)/(dx) \right)-\left(-4y^(-5)(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+(4)/(y^5)}=\cfrac{dy}{dx} \right|_(3,1)\implies \cfrac{1}{20(1)^4+(4)/((1)^5)}\implies \cfrac{1}{24}`