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Chapter 13: Problem 31

Find the four second partial derivatives of the following functions. $$p(u, v)=\ln \left(u^{2}+v^{2}+4\right)$$

Short Answer

Expert verified
Question: Calculate the four second partial derivatives of the function \(p(u, v) = \ln(u^2 + v^2 + 4)\). Answer: The four second partial derivatives are: 1. \(\frac{\partial^2 p(u, v)}{\partial u^2} = \frac{2\left(v^{2}-u^{2}+4\right)}{(u^{2}+v^{2}+4)^{2}}\) 2. \(\frac{\partial^2 p(u, v)}{\partial u \partial v} = -\frac{4uv}{(u^{2}+v^{2}+4)^{2}}\) 3. \(\frac{\partial^2 p(u, v)}{\partial v \partial u} = -\frac{4uv}{(u^{2}+v^{2}+4)^{2}}\) 4. \(\frac{\partial^2 p(u, v)}{\partial v^2} = \frac{2\left(u^{2}-v^{2}+4\right)}{(u^{2}+v^{2}+4)^{2}}\)

Step by step solution

01

Find the first partial derivatives

Compute the first partial derivatives of \(p(u, v)=\ln \left(u^{2}+v^{2}+4\right)\) with respect to \(u\) and \(v\): $$\frac{\partial p(u, v)}{\partial u} = \frac{\partial}{\partial u}\ln \left(u^{2}+v^{2}+4\right) = \frac{2u}{u^{2}+v^{2}+4}$$ $$\frac{\partial p(u, v)}{\partial v} = \frac{\partial}{\partial v}\ln \left(u^{2}+v^{2}+4\right) = \frac{2v}{u^{2}+v^{2}+4}$$
02

Find the second partial derivative with respect to \(u\) twice

Differentiate the first partial derivative with respect to \(u\) again to find the second partial derivative: $$\frac{\partial^2 p(u, v)}{\partial u^2} = \frac{\partial}{\partial u} \frac{2u}{u^{2}+v^{2}+4} = \frac{2\left(u^{2}+v^{2}+4\right) - 2u(2u)}{(u^{2}+v^{2}+4)^{2}} = \frac{2\left(v^{2}-u^{2}+4\right)}{(u^{2}+v^{2}+4)^{2}}$$
03

Find the second mixed partial derivative with respect to \(u\) and \(v\)

Differentiate the first partial derivative with respect to \(u\) once and then with respect to \(v\): $$\frac{\partial^2 p(u, v)}{\partial u \partial v} = \frac{\partial}{\partial v} \frac{2u}{u^{2}+v^{2}+4} = \frac{-2u(2v)}{(u^{2}+v^{2}+4)^{2}} = -\frac{4uv}{(u^{2}+v^{2}+4)^{2}}$$
04

Find the second mixed partial derivative with respect to \(v\) and \(u\)

Differentiate the first partial derivative with respect to \(v\) once and then with respect to \(u\): $$\frac{\partial^2 p(u, v)}{\partial v \partial u} = \frac{\partial}{\partial u} \frac{2v}{u^{2}+v^{2}+4} = \frac{-2v(2u)}{(u^{2}+v^{2}+4)^{2}} = -\frac{4uv}{(u^{2}+v^{2}+4)^{2}}$$
05

Find the second partial derivative with respect to \(v\) twice

Differentiate the first partial derivative with respect to \(v\) again to find the second partial derivative: $$\frac{\partial^2 p(u, v)}{\partial v^2} = \frac{\partial}{\partial v} \frac{2v}{u^{2}+v^{2}+4} = \frac{2\left(u^{2}+v^{2}+4\right) - 2v(2v)}{(u^{2}+v^{2}+4)^{2}} = \frac{2\left(u^{2}-v^{2}+4\right)}{(u^{2}+v^{2}+4)^{2}}$$ The four second partial derivatives are: 1. \(\frac{\partial^2 p(u, v)}{\partial u^2} = \frac{2\left(v^{2}-u^{2}+4\right)}{(u^{2}+v^{2}+4)^{2}}\) 2. \(\frac{\partial^2 p(u, v)}{\partial u \partial v} = -\frac{4uv}{(u^{2}+v^{2}+4)^{2}}\) 3. \(\frac{\partial^2 p(u, v)}{\partial v \partial u} = -\frac{4uv}{(u^{2}+v^{2}+4)^{2}}\) 4. \(\frac{\partial^2 p(u, v)}{\partial v^2} = \frac{2\left(u^{2}-v^{2}+4\right)}{(u^{2}+v^{2}+4)^{2}}\)

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Most popular questions from this chapter

Show that $$\lim _{(x, y) \rightarrow(0,0)} \frac{a x^{2(p-n)} y^{n}}{b x^{2 p}+c y^{p}} \text { does }$$ not exist when \(a, b,\) and \(c\) are nonzero real numbers and \(n\) and \(p\) are positive integers with \(p \geq n\)

The domain of $$f(x, y)=e^{-1 /\left(x^{2}+y^{2}\right)}$$ excludes \((0,0) .\) How should \(f\) be defined at (0,0) to make it continuous there?

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x^{2}+y^{2}}}$$

Absolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=x^{2}+4 y^{2}+1 ; R=\left\\{(x, y): x^{2}+4 y^{2} \leq 1\right\\}$$

Find the points (if they exist) at which the following planes and curves intersect. $$8 x+15 y+3 z=20 ; \quad \mathbf{r}(t)=\langle 1, \sqrt{t},-t\rangle, \text { for } t>0$$
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