91Ó°ÊÓ

Find the first partial derivatives of the function. $$z=\tan x y$$

Find the directional derivative of the function at the given point in the direction of the vector \(v\) . \(g(r, s)=\tan ^{-1}(r s), \quad(1,2), \quad \mathbf{v}=[5,10]\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. \(f(x, y)=e^{y}\left(y^{2}-x^{2}\right)\)

Given that \(f\) is a differentiable function with \(f(2,5)=6\) , \(f_{x}(2,5)=1,\) and \(f_{y}(2,5)=-1,\) use a linear approximation to estimate \(f(2.2,4.9)\)

Find and sketch the domain of the function. \(f(x, y)=\sqrt{1-x^{2}}-\sqrt{1-y^{2}}\)

Find the first partial derivatives of the function. $$f(x, y)=\frac{x-y}{x+y}$$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. \(f(x, y)=y^{2}-2 y \cos x, \quad-1 \leqslant x \leqslant 7\)

Find the directional derivative of the function at the given point in the direction of the vector \(v\) . \(V(u, t)=e^{-u t}, \quad(0,3), \quad \mathbf{v}=[2,-1]\)

Find the first partial derivatives of the function. $$f(x, y)=x^{y}$$

Find and sketch the domain of the function. \(f(x, y)=\ln \left(x^{2}+y^{2}-2\right)\)