If z = e^xsiny, x = log_et and y = t² then \(\frac{{dz}}{{dt}}\) is given by the expression

Concept:

If z = f(x, y) and x, y is the function of t.

Then \(\frac{dz}{dt}=\frac{\partial z}{\partial x}\times\frac{dx}{dt}\;+\;\frac{\partial z}{\partial y}\times\frac{dy}{dt}\)

Calculation:

Given:

z = e^xsiny ⇒ \(\frac{{\partial z}}{{\partial x}} = {e^x}siny\;\;and\;\;\frac{{\partial z}}{{\partial y}} = {e^x}cosy\)

x = log_et ⇒ \(\frac{{dx}}{{dt}} = \frac{1}{t}\)

y = t² ⇒ \(\frac{{dy}}{{dt}} = 2t\)

\(\frac{{dz}}{{dt}} = \frac{{\partial z}}{{\partial x}}\times\frac{{dx}}{{dt}} + \frac{{\partial z}}{{\partial y}}\times\frac{{dy}}{{dt}}\)

\(\therefore\frac{dz}{dt}=\left (\frac{e^xsin\;y}{t} \right )\;+\;2te^xcos\;y\)

\(\therefore\;\frac{{{e^x}}}{t}(siny + 2{t^2}cosy)\)