Question

Suppose w=xy+yzw=xy+yz, where x=e2t, y=2+sin(4t)x=e2t, y=2+sin⁡(4t), and z=2+cos(5t)z=2+cos⁡(5t).

Answer 1

$w(x,y,z)=xy+yz$
$x(t)=e^{2t}$
$y(t)=2+\sin4t$
$z(t)=2+\cos5t$

$\dfrac{\mathrm dw}{\mathrm dt}=\dfrac{\partial w}{\partial x}\dfrac{\mathrm dx}{\mathrm dt}+\dfrac{\partial w}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}+\dfrac{\partial w}{\partial z}\dfrac{\mathrm dz}{\mathrm dt}$

$\dfrac{\mathrm dw}{\mathrm dt}=y\dfrac{\mathrm dx}{\mathrm dt}+(x+z)\dfrac{\mathrm dy}{\mathrm dt}+y\dfrac{\mathrm dz}{\mathrm dt}$

$\dfrac{\mathrm dw}{\mathrm dt}=2ye^{2t}+4(x+z)\cos4t-5y\sin5t$