Question

Use the chain rule to find the indicated partial derivatives. z = x4 + x2y, x = s + 2t − u, y = stu2; ∂z ∂s , ∂z ∂t , ∂z ∂u when s = 3, t = 1, u = 2

Answer 1

By the chain rule,

$\dfrac{\partial z}{\partial s}=\dfrac{\partial z}{\partial x}\cdot\dfrac{\partial x}{\partial s}+\dfrac{\partial z}{\partial y}\cdot\dfrac{\partial y}{\partial s}$

$\dfrac{\partial z}{\partial s}=(4x^3+2xy)(1)+(x^2)(tu^2)$

When $(s,t,u)=(3,1,2)$, we get $(x,y)=(3,12)$, so at this point the partial derivative of $z$ with respect to $s$ takes on a value of

$\dfrac{\partial z}{\partial s}\bigg|_{s=3,t=1,u=2}=216$