In deriving the work-energy theorem , we used the chain rule to Calculate the derivative $d E / d t$. This is actually a rather subtle point because $E$ is a function of three variables, $E\left(p_x, p_y, p_z\right)$, and the relevant form of the chain rule is

$$d E=\frac{\partial E}{\partial p_x} d p_x+\frac{\partial E}{\partial p_y} d p_y+\frac{\partial E}{\partial p_z} d p_z$$

where the three derivatives are partial derivatives. Use this equation to fill in the details.

Making Predictions How might economically developing nations view the cold war?

For the composite channel section shown in Fig., Determine the critical depth $y_{c}$ if (a) $Q=55 \mathrm{~m}^{3} / \mathrm{s}$ (b) $Q=3.5 \mathrm{~m}^{3} / \mathrm{s}$

Discuss what happens to $y$ as a function of $x$ if $\partial z / \partial y=0$ in (1) for the given equation. $x-\sin y=0$

Acetylene gas $\text{(C}_2\text{H}_2)$ is often used by plumbers, welders, and glass blowers because it burns in oxygen with an intensely hot flame. The products of the combustion of acetylene are carbon dioxide and water vapor. Write the unbalanced chemical equation for this process.

Use multiplication to solve the proportion.

9/5 = z/20

Solve the system

u = 3x + 2y, v = x + 4y

for $x$ and $y$ in terms of $u$ and $v$. Then find the value of the Jacobian $\partial(x, y) / \partial(u, v)$.

Summarize the characteristics shared by mollusks and annelids.

A certain $p$-channel JFET has a $V_{G S(o f)}=6 \mathrm{~V}$. What is $I_{\mathrm{D}}$ when $V_{G S}=8 \mathrm{~V}$ ?

Discuss what happens to $y$ as a function of $x$ if $\partial z / \partial y=0$ in (1) for the given equation. $x-y^2=0$

Find the partial derivatives. Assume the variables are restricted to a domain on which the function is defined.

$\frac{\partial T}{\partial l}$ if $T=2 \pi \sqrt{\frac{l}{g}}$

Use the Chain Rule to calculate the partial derivatives. Express the answer in terms of the independent variables.

$$\frac { \partial f } { \partial \theta } ; f ( x , y , z ) = x y - z ^ { 2 } , x = r \cos \theta , y = \cos ^ { 2 } \theta,$$

z = r

In the following exercise, determine the partial derivative with respect to x and the partial derivative with respect to y.

$$f(x, y)=\sqrt[3]{x^2 y}$$

Write the rational function given as a partial sum of partial fraction.

$$\frac{1}{x^{3}-1}$$