Answer to Question #136785 in Calculus for Promise Omiponle

Question

Suppose z=x^2siny, x=−4s^2+t^2, y=−2st.

A. Use the chain rule to find ∂z/∂s and ∂z/∂t as functions of x, y, s and t.
∂z/∂s=

∂z/∂t=

B. Find the numerical values of ∂z/∂s and ∂z/∂t when (s,t)=(−5,−5).
∂z/∂s(−5,−5)=

∂z/∂t(−5,−5)=

Answer 1

$\displaystyle z=x^2\sin y, x=−4s^2+t^2, y=−2st.\\ A. \\\textsf{By Chain rule or composite rule of differentiation,}\\ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\times \frac{\partial x}{\partial s}\\ \frac{\partial z}{\partial x} = 2x\sin{y}\, \frac{\partial x}{\partial s} = -8s\\ \therefore \frac{\partial z}{\partial x} = -16xs\sin{y} \\ B.\\ \begin{aligned} \frac{\partial z}{\partial t} &= \frac{\partial z}{\partial x} \times \frac{\partial x}{\partial t} \\&= -16xs\sin{y} \times 2t = -32xst\sin{y} \end{aligned}\\ \textsf{At}\, (s, t) = (-5, -5)\\ y = (-2)(-5)(-5) = -50\\ x = -4(-5)^2 + (-5)^2 = -100 + 25 = -75\\ \therefore \frac{\partial z}{\partial x} = -16(-75)(-5)\sin{(-50)} = 6000\sin(50) = -1574.249\,\&\\ \frac{\partial z}{\partial t} = 2(-5)\times 6000\sin(50) = -60000\sin(50)=15742.49$