Chapter 12: Q. 24 (page 989)

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y, z) = \ln (x + y + z)$

Short Answer

Expert verified

$\frac{\partial^{2} f}{\partial x^{2}} = - \frac{1}{(x + y + z)^{2}}$

$\frac{\partial^{2} f}{\partial y^{2}} = - \frac{1}{(x + y + z)^{2}}$

$\frac{\partial^{2} f}{\partial z^{2}} = - \frac{1}{(x + y + z)^{2}}$

Step by step solution

Step 1. Given information

$f (x, y, z) = \ln (x + y + z)$

Step 2. Differentiate

Partially differentiating with respect to x,y and z,

$\begin{aligned} \frac{\partial f}{\partial x} = \frac{1}{x + y + z} \\ \frac{\partial f}{\partial y} = \frac{1}{x + y + z} \\ \frac{\partial f}{\partial z} = \frac{1}{x + y + z} \end{aligned}$

Again differentiating with respect to x, y and z,

$\begin{aligned} \frac{\partial^{2} f}{\partial x^{2}} = - \frac{1}{(x + y + z)^{2}} \\ \frac{\partial^{2} f}{\partial z \partial x} = - \frac{1}{(x + y + z)^{2}} = \frac{\partial^{2} f}{\partial x \partial z} \\ \frac{\partial^{2} f}{\partial y \partial x} = - \frac{1}{(x + y + z)^{2}} = \frac{\partial^{2} f}{\partial x \partial y} \end{aligned}$

Again differentiating with respect to y and z,

$\begin{aligned} \frac{\partial^{2} f}{\partial y^{2}} = - \frac{1}{(x + y + z)^{2}} \\ \frac{\partial^{2} f}{\partial z \partial y} = - \frac{1}{(x + y + z)^{2}} = \frac{\partial^{2} f}{\partial y \partial z} \end{aligned}$

Now,

$\frac{\partial^{2} f}{\partial z^{2}} = - \frac{1}{(x + y + z)^{2}}$

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Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y, z) = \ln (x + y + z)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Differentiate

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Partial derivatives: Find all first- and second-order partial derivatives for the following functions:f(x,y,z)=ln⁡(x+y+z)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Differentiate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Discrete Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:
$f (x, y, z) = \ln (x + y + z)$