91Ó°ÊÓ

Find the gradient of the function. Assume the variables are restricted to a domain on which the function s defined. $$f(r, \theta)=r \sin \theta$$

Find the partial derivatives. The variables are restricted to a domain on which the function is defined. $$z_{x} \text { if } z=x^{2} y+2 x^{5} y$$

Find the gradient of the function. $$f(x, y, z)=x y+\sin \left(e^{z}\right)$$

List the points in the \(x y\) -plane, if any, at which the function \(z=f(x, y)\) is not differentiable. $$z=4+\sqrt{(x-1)^{2}+(y-2)^{2}}$$

find the differential of the function. $$f(x, y)=\sin (x y)$$

Find \(\partial z / \partial u\) and \(\partial z / \partial v .\) The variables are restricted to domains on which the functions are defined. $$z=x e^{y}, x=\ln u, y=v$$

Calculate all four second-order partial derivatives and check that \(f_{x y}=f_{y x} .\) Assume the variables are restricted to a domain on which the function is defined. $$f(x, y)=5 x^{3} y^{2}-7 x y^{3}+9 x^{2}+11$$

The quantity, \(Q\), of beef purchased at a store, in kilograms per week, is a function of the price of beef, \(b,\) and the price of chicken, \(c,\) both in dollars per kilogram. (a) Do you expect \(\partial Q / \partial b\) to be positive or negative? Explain. (b) Do you expect \(\partial Q / \partial c\) to be positive or negative? Explain. (c) Interpret the statement \(\partial Q / \partial b=-213\) in terms of quantity of beef purchased.

Find the partial derivatives. The variables are restricted to a domain on which the function is defined. $$\frac{\partial}{\partial x}(a \sqrt{x})$$

find the differential of the function. $$g(u, v)=u^{2}+u v$$