Question

Utilize a regra da cadeia e determine dz/dt de z=tg(x^2 + y), com x=2t e y= t^2?

Answer 1

$\displaystyle z(x,y)=\tan(x^2+y)\\x=2t\\y=t^2\\z(t)=\tan(4t^2+t^2)=\tan(5t^2) \\\tan(u)=z(u)\longrightarrow\ u(t)=5t^2 \\\\ \frac{dz}{dt}=\frac{dz}{du}\cdot\frac{du}{dt}\\\\ \frac{d}{dt}\tan(5t^2)=\frac{d}{du}\tan(u)\cdot\frac{d}{dt}5t^2=\sec^2(u)\cdot10t\rightarrow u=5t^2\\\\ \boxed{\frac{d}{dt}\tan(5t^2)=10t\sec^2(5t^2)}$