[tex]\frac{d}{dx}[f(g(x))]=\frac{d}{d[g(x)]}[f(g(x))].\frac{d}{dx}[g(x)][/tex]
[tex]\frac{d}{dx}(x^n)=n.x^{n-1}[/tex]
Nesse exercício:
[tex]f(g(x))=(64+4t)^{\frac{2}{3}}[/tex]
[tex]g(x)=64+4t=u[/tex]
[tex]f(u)=u^{\frac{2}{3}}[/tex]
[tex]N(t)=100(64+4t)^{\frac{2}{3}}\\\\N'(t)=\frac{2}{3}.100(64+4t)^{\frac{2}{3}-\frac{3}{3}}.(64+4t)'\\\\N'(t)=\frac{2}{3}.100(64+4t)^{\frac{-1}{3}}.4\\\\N'(t)=\frac{800}{3.\sqrt[3]{64+4t}}\\\\N'(0)=\frac{800}{3\sqrt[3]{64}}\\\\N'(0)=\frac{200}{3}[/tex]
