14.3 Videos Guide

14.3a

Definition: (partial derivative)

• $\frac{\partial z}{\partial x}=f_{x}(x,\ y)=\lim_{h\rightarrow0}\frac{f\left(x+h,\text{ }y\right)-f\left(x,\text{ }y\right)}{h}$

• $\frac{\partial z}{\partial y}=f_{y}(x,\ y)=\lim_{h\rightarrow0}\frac{f\left(x,\text{ }y+h\right)-f\left(x,\text{ }y\right)}{h}$

Exercise:

• Find the first partial derivatives of the function.
  $z=x\sin{(xy)}$

14.3b

• Higher-order partial derivatives

  • $\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial^{2}y}{\partial y\partial x}=f_{xy}$ denotes the partial derivative first with respect to $x$ and then with respect to $y$

Theorem (statement):

• The conclusion of Clairaut’s Theorem is that second mixed partial derivatives are equal:
  $f_{xy}=f_{yx}$

Exercises:

• Verify that the conclusion of Clairaut’s Theorem holds, that is, $u_{xy}=u_{yx}$.
  $u=e^{xy}\sin y$

14.3c

• Find the first partial derivatives of the function.
  $f(x,\ y,\ z)=xy^{2}e^{-xz}$

• Use implicit differentiation to find $\frac{\partial x}{\partial z}$ and $\frac{\partial x}{\partial y}$.
  $x^{2}-y^{2}+z^{2}-2z=4$

14.3d

• Determine the signs of the partial derivatives for the function $f$ whose graph is shown.

(a) $f_{xy}(1,\ 2)$ (b) $f_{xy}(-1,\ 2)$