Question

Pls explain answer c and d?

Answer 1

$\bf a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} \qquad \qquad \sqrt[ m]{a^ n}\implies a^{\frac{ n}{ m}}\\\\ -------------------------------\\\\ c)\\\\ f(x)=\cfrac{1}{\sqrt{x}}\implies f(x)=x^{-\frac{1}{2}}\implies \cfrac{dy}{dx}=-\cfrac{1}{2}x^{-\frac{1}{2}-1}\implies \cfrac{dy}{dx}=-\cfrac{1}{2}x^{-\frac{3}{2}}$

$\bf \cfrac{dy}{dx}=-\cfrac{1}{2}\cdot \cfrac{1}{x^{\frac{3}{2}}}\implies \cfrac{dy}{dx}=-\cfrac{1}{2}\cdot\cfrac{1}{\sqrt[2]{x^3}}\implies \cfrac{dy}{dx}=-\cfrac{1}{2\sqrt{x^3}}\\\\ -------------------------------\\\\ d)\\\\ f(t)=t^3+3t^2-6\implies \cfrac{dy}{dt}=3t^2+6t^1-0\implies \cfrac{dy}{dt}=3t^2+6t$