91Ó°ÊÓ

Find the derivative of the function at \(P_{0}\) in the direction of \(\mathbf{u}.\) \(\begin{array}{c}g(x, y)=\frac{x-y}{x y+2}, \quad P_{0}(1,-1), \quad \mathbf{u}=12 \mathbf{i}+5 \mathbf{j}\end{array}\)

In Exercises \(1-22,\) find \(\partial f / \partial x\) and \(\partial f / \partial y\) $$f(x, y)=e^{(x+y+1)}$$

In Exercises \(13-24\) , draw a branch diagram and write a Chain Rule formula for each derivative. \(\frac{d z}{d t} \text { for } z=f(x, y), \quad x=g(t), \quad y=h(t)\)

In Exercises \(13-16,\) find and sketch the level curves \(f(x, y)=c\) on the same set of coordinate axes for the given values of \(c .\) We refer to these level curves as a contour map. $$ f(x, y)=x+y-1, \quad c=-3,-2,-1,0,1,2,3 $$

Find all the local maxima, local minima, and saddle points of the functions. $$ f(x, y)=x^{3}-y^{3}-2 x y+6 $$

Find all the local maxima, local minima, and saddle points of the functions. $$ f(x, y)=x^{3}+3 x y+y^{3} $$

Find the derivative of the function at \(P_{0}\) in the direction of \(\mathbf{u}.\) \(\begin{array}{ll}{h(x, y)=\tan ^{-1}(y / x)+\sqrt{3} \sin ^{-1}(x y / 2),} & {P_{0}(1,1)}, \\ {\mathbf{u}=3 \mathbf{i}-2 \mathbf{j}}\end{array}\)

Draw a branch diagram and write a Chain Rule formula for each derivative. \(\frac{d z}{d t} \text { for } z=f(u, v, w), \quad u=g(t), \quad v=h(t), \quad w=k(t)\)

In Exercises \(1-22,\) find \(\partial f / \partial x\) and \(\partial f / \partial y\) $$f(x, y)=e^{-x} \sin (x+y)$$

In Exercises \(13-16,\) find and sketch the level curves \(f(x, y)=c\) on the same set of coordinate axes for the given values of \(c .\) We refer to these level curves as a contour map. $$ f(x, y)=x^{2}+y^{2}, \quad c=0,1,4,9,16,25 $$