Question

Найдите производную сложной функции: z=xe^y, где x=5t^2 -t, y=6t

Answer 1

$z=xe^{y}\; ,\; \; x=5t^2-t\; ,\; \; y=6t\\\\\boxed {z=f(x,y)\; ,\; \; x=x(t)\; ,\; y=y(t)\; \; \to \; \; \frac{dz}{dt}=\frac{\partial z}{\partial x}\cdot \frac{dx}{dt}+\frac{\partial z}{\partial y}\cdot \frac{dy}{dt}}\\\\\\\frac{dz}{dt}=e^{y}\cdot (10t-1)+x\cdot e^{y}\cdot 6=e^{y}\cdot (10t-1+6x)=\\\\=e^{6t}\cdot (10t-1+30t^2-6t)=e^{6t}\cdot (30t^2+4t-1)$