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).a. use the chain rule to find dwdtdwdt as a function of x, y, z, and t. do not rewrite x, y, and z in terms of t, and do not rewrite e2te2t as x.

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

[tex]\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}[/tex]

[tex]\dfrac{\mathrm dw}{\mathrm dt}=y\dfrac{\mathrm dx}{\mathrm dt}+(x+z)\dfrac{\mathrm dy}{\mathrm dt}+y\dfrac{\mathrm dz}{\mathrm dt}[/tex]

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