Chapter 4

Find \(\frac{d y}{d x}\) using partial derivatives. \(x \cos (x y)+y \cos x=2\)

Find \(\frac{d y}{d x}\) using partial derivatives. \(x^{2} y^{3}+\cos y=0\)

Find \(\frac{d z}{d t}\) using the chain rule where \(z=3 x^{2} y^{3}, x=t^{4},\) and \(y=t^{2}\)

Let \(z=3 \cos x-\sin (x y), x=\frac{1}{t}, \quad\) and \(\quad y=3 t\). Find \(\frac{d z}{d t}\).

Let \(z=e^{1-x y}, x=t^{1 / 3},\) and \(y=t^{3} .\) Find \(\frac{d z}{d t}\)

Find \(\frac{d z}{d t}\) by the chain rule where \(z=\cosh ^{2}(x y), x=\frac{1}{2} t,\) and \(y=e^{t}\)

. Let \(z=\frac{x}{y}, x=2 \cos u,\) and \(y=3 \sin v\). Find \(\frac{\partial z}{\partial u}\) and \(\frac{\partial z}{\partial v}\).

Let \(z=e^{x^{2} y},\) where \(x=\sqrt{u v}\) and \(y=\frac{1}{v} .\) Find \(\frac{\partial z}{\partial u}\) and \(\frac{\partial z}{\partial v}\)

If \(z=x y e^{x / y}, x=r \cos \theta,\) and \(y=r \sin \theta,\) find \(\frac{\partial z}{\partial r}\) and \(\frac{\partial z}{\partial \theta}\) when \(r=2\) and \(\theta=\frac{\pi}{6}\).

Find \(\quad \frac{\partial w}{\partial s}\) \(w=4 x+y^{2}+z^{3}, x=e^{r s^{2}}, y=\ln \left(\frac{r+s}{t}\right),\) \(z=r s t^{2}\).

If \(\quad w=\sin (x y z), x=1-3 t, y=e^{1-t}, \quad\) and \(z=4 t,\) find \(\frac{\partial w}{\partial t} .\)

Use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y) .\) Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=3 x^{2}+y^{2}\)

Use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y) .\) Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=\sqrt{x^{2}+y^{2}}\)

Use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y) .\) Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\). \(f(x, y)=x^{2} y-2 y^{3}\)

The volume of a right circular cylinder is given by \(V(x, y)=\pi x^{2} y,\) where \(x\) is the radius of the cylinder and \(y\) is the cylinder height. Suppose \(x\) and \(y\) are functions of \(t\) given by \(x=\frac{1}{2} t\) and \(y=\frac{1}{3} t\) so that \(x\) and \(y\) are both increasing with time. How fast is the volume increasing when \(x=2\) and \(y=5 ?\)

The radius of a right circular cone is increasing at 3 \(\mathrm{cm} / \mathrm{min}\) whereas the height of the cone is decreasing at 2 \(\mathrm{cm} / \mathrm{min}\). Find the rate of change of the volume of the cone when the radius is \(13 \mathrm{~cm}\) and the height is \(18 \mathrm{~cm}\).

A closed box is in the shape of a rectangular solid with dimensions \(x, y,\) and \(z .\) (Dimensions are in inches.) Suppose each dimension is changing at the rate of 0.5 in./min. Find the rate of change of the total surface area of the box when \(x=2\) in., \(y=3\) in., and \(z=1\) in.

The total resistance in a circuit that has three individual resistances represented by \(x, y,\) and \(z\) is given by the formula \(R(x, y, z)=\frac{x y z}{y z+x z+x y} .\) Suppose at a given time the \(x\) resistance is \(100 \Omega\), the \(y\) resistance is \(200 \Omega,\) and the \(z\) resistance is \(300 \Omega\). Also, suppose the \(x\) resistance is changing at a rate of \(2 \Omega / \mathrm{min}\), the \(y\) resistance is changing at the rate of \(1 \Omega / \mathrm{min},\) and the \(z\) resistance has no change. Find the rate of change of the total resistance in this circuit at this time.

The temperature \(T\) at a point \((x, y)\) is \(T(x, y)\) and is measured using the Celsius scale. A fly crawls so that its position after \(t\) seconds is given by \(x=\sqrt{1+t}\) and \(y=2+\frac{1}{3} t,\) where \(x\) and \(y\) are measured in centimeters. The temperature function satisfies \(T_{x}(2,3)=4\) and \(T_{y}(2,3)=3\). How fast is the temperature increasing on the fly's path after 3 sec?

The \(x\) and \(y\) components of a fluid moving in two dimensions are given by the following functions: \(u(x, y)=2 y\) and \(v(x, y)=-2 x ; \quad x \geq 0 ; y \geq 0 .\) The speed of the fluid at the point \((x, y)\) is \(s(x, y)=\sqrt{u(x, y)^{2}+v(x, y)^{2}}\). Find \(\frac{\partial s}{\partial x}\) and \(\frac{\partial s}{\partial y}\) using the chain rule.

\(\quad\) Let \(\quad u=u(x, y, z),\) where \(x=x(w, t), y=y(w, t), z=z(w, t), w=w(r, s),\) and \(t=t(r, s) .\) Use a tree diagram and the chain rule to find an expression for \(\frac{\partial u}{\partial r}\).

For the following exercises, find the directional derivative using the limit definition only. \( f(x, y)=5-2 x^{2}-\frac{1}{2} y^{2}\) at point \(P(3,4)\) in the direction of \(\mathrm{u}=\left(\cos \frac{\pi}{4}\right) \mathrm{i}+\left(\sin \frac{\pi}{4}\right) \mathrm{j}\).

For the following exercises, find the directional derivative using the limit definition only. \( f(x, y)=y^{2} \cos (2 x)\) at point \(P\left(\frac{\pi}{3}, 2\right)\) in the direction of \(\mathrm{u}=\left(\cos \frac{\pi}{4}\right) \mathrm{i}+\left(\sin \frac{\pi}{4}\right) \mathrm{j}\).

For the following exercises, find the directional derivative using the limit definition only. Find the directional derivative of \(f(x, y)=y^{2} \sin (2 x)\) at point \(P\left(\frac{\pi}{4}, 2\right)\) in the direction of \(\mathbf{u}=5 \mathbf{i}+12 \mathbf{j}\).

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=x y, \quad P(0,-2), \quad \mathrm{v}=\frac{1}{2} \mathbf{i}+\frac{\sqrt{3}}{2} \mathbf{j}$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$h(x, y)=e^{x} \sin y, P\left(1, \frac{\pi}{2}\right), \mathbf{v}=-\mathbf{i}$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$h(x, y, z)=x y z, P(2,1,1), \mathrm{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k}$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=x y, P(1,1), \mathbf{u}=\left\langle\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right\rangle$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=x^{2}-y^{2}, \mathbf{u}=\left\langle\frac{\sqrt{3}}{2}, \frac{1}{2}\right\rangle, P(1,0)$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=3 x+4 y+7, \mathbf{u}=\left\langle\frac{3}{5}, \frac{4}{5}\right\rangle, P\left(0, \frac{\pi}{2}\right)$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=e^{x} \cos y, \quad \mathbf{u}=\langle 0,1\rangle, \quad P=\left(0, \frac{\pi}{2}\right)$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=y^{10}, \quad \mathbf{u}=\langle 0,-1\rangle, \quad P=(1,-1)$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=\ln \left(x^{2}+y^{2}\right), \mathbf{u}=\left\langle\frac{3}{5}, \frac{4}{5}\right\rangle, \quad P(1,2)$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=x^{2} y, P(-5,5), \quad \mathbf{v}=3 \mathbf{i}-4 \mathbf{j}$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(\mathbf{v}\). $$f(x, y)=y^{2}+x z, P(1,2,2), \quad \mathbf{v}=\langle 2,-1,2\rangle$$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). $$f(x, y)=x^{2}+2 y^{2}, \theta=\frac{\pi}{6}$$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). $$f(x, y)=\frac{y}{x+2 y}, \theta=-\frac{\pi}{4}$$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). $$f(x, y)=\cos (3 x+y), \theta=\frac{\pi}{4}$$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). $$w(x, y)=y e^{x}, \theta=\frac{\pi}{3}$$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). $$f(x, y)=x \arctan (y), \quad \theta=\frac{\pi}{2}$$

For the following exercises, find the directional derivative of the function in the direction of the unit vector \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). $$f(x, y)=\ln (x+2 y), \quad \theta=\frac{\pi}{3}$$

For the following exercises, find the gradient. Find the gradient of \(f(x, y)=\frac{14-x^{2}-y^{2}}{3}\). Then, find the gradient at point \(P(1,2)\).

For the following exercises, find the gradient. Find the gradient of \(f(x, y, z)=x y+y z+x z\) at point \(P(1,2,3)\).

For the following exercises, find the gradient. Find the gradient of \(f(x, y, z)\) at \(P\) and in the direction \(\quad\) of \(f(x, y, z)=\ln \left(x^{2}+2 y^{2}+3 z^{2}\right), P(2,1,4), \quad \mathbf{u}=\frac{-3}{13} \mathbf{i}-\frac{4}{13} \mathbf{j}-\frac{12}{13} \mathbf{k}\).

For the following exercises, find the gradient. \(f(x, y, z)=4 x^{5} y^{2} z^{3}, P(2,-1,1), \quad \mathbf{u}=\frac{1}{3} \mathbf{i}+\frac{2}{3} \mathbf{j}-\frac{2}{3} \mathbf{k}\).

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(Q\). $$ f(x, y)=x^{2}+3 y^{2}, P(1,1), \quad Q(4,5)$$

For the following exercises, find the directional derivative of the function at point \(P\) in the direction of \(Q\). $$ f(x, y, z)=\frac{y}{x+z}, P(2,1,-1), \quad Q(-1,2,0)$$

For the following exercises, find the derivative of the function at \(P\) in the direction of \(\mathbf{u}\). $$ f(x, y)=-7 x+2 y, P(2,-4), \quad \mathbf{u}=4 \mathbf{i}-3 \mathbf{j}$$

For the following exercises, find the derivative of the function at \(P\) in the direction of \(\mathbf{u}\). $$ f(x, y)=\ln (5 x+4 y), P(3,9), \quad \mathbf{u}=6 \mathbf{i}+8 \mathbf{j}$$

[T] Use technology to sketch the level curve of \(f(x, y)=4 x-2 y+3\) that passes through \(P(1,2)\) and draw the gradient vector at \(P\).