Example 4-3-25 - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Education : Study Guides : MultivariateCalculus : Chapter 4 : Examples : Section 4-3 : Example 4-3-25

Chapter 4: Partial Differentiation

Section 4.3: Chain Rule

Example 4.3.25

If the equation $x^{2} + y^{3} + z^{4} + u^{5} = 1$ implicitly defines $u = u (x, y, z)$ , and the equation $x + y^{2} + z^{3} = 1$ implicitly defines $z = z (x, y)$ , obtain $u_{x}$ and $u_{y}$ .

Solution

Mathematical Solution

Differentiate each equation with respect to $x$ , keeping in mind the implicitly defined functions in each.

$\frac{\partial}{\partial x} (x^{2} + y^{3} + z^{4} + u^{5} = 1) \Rightarrow 2 x + 4 z^{3} z_{x} + 5 u^{4} u_{x} = 0 \Rightarrow u_{x} = - \frac{2 x + 4 z^{3} z_{x}}{5 u^{4}}$

and

$\frac{\partial}{\partial x} (x + y^{2} + z^{3} = 1) \Rightarrow 1 + 3 z^{2} z_{x} = 0 \Rightarrow z_{x} = - \frac{1}{3 z^{2}}$

from which it follows that $u_{x} = - \frac{2 x + 4 z^{3} (- \frac{1}{3 z^{2}})}{5 u^{4}} = \frac{4 z - 6 x}{15 u^{4}}$ .

Differentiate each equation with respect to $y$ , keeping in mind the implicitly defined functions in each.

$\frac{\partial}{\partial y} (x^{2} + y^{3} + z^{4} + u^{5} = 1) \Rightarrow 3 y^{2} + 4 z^{3} z_{y} + 5 u^{4} u_{y} = 0 \Rightarrow u_{y} = - \frac{3 y^{2} + 4 z^{3} z_{y}}{5 u^{4}}$

and

$\frac{\partial}{\partial y} (x + y^{2} + z^{3} = 1) \Rightarrow 2 y + 3 z^{2} z_{y} = 0 \Rightarrow z_{y} = - \frac{2 y}{3 z^{2}}$

from which it follows that $u_{y} = - \frac{3 y^{2} + 4 z^{3} (- \frac{2 y}{3 z^{2}})}{5 u^{4}} = \frac{8 y z - 9 y^{2}}{15 u^{4}}$ .

Maple Solution - Interactive

Obtain $u_{x}$ from first principles

•	Write the first equation with the appropriate dependencies made explicit.

•	Context Panel: Differentiate≻With Respect To≻ $x$

$x^{2} + y^{3} + z {(x, y)}^{4} + u {(x, y, z)}^{5} = 1$

$x^{2} + y^{3} + {z (x, y)}^{4} + {u (x, y, z)}^{5} = 1$

(1)

$\overset{differentiate w.r.t. x}{\to}$

$2 x + 4 {z (x, y)}^{3} (\frac{\partial}{\partial x} z (x, y)) + 5 {u (x, y, z)}^{4} (\frac{\partial}{\partial x} u (x, y, z)) = 0$

(2)

•	Write the second equation with the appropriate dependencies made explicit.

•	Context Panel: Differentiate≻With Respect To≻ $x$

$x + y^{2} + z {(x, y)}^{3} = 1$

$x + y^{2} + {z (x, y)}^{3} = 1$

(3)

$\overset{differentiate w.r.t. x}{\to}$

$1 + 3 {z (x, y)}^{2} (\frac{\partial}{\partial x} z (x, y)) = 0$

(4)

•	Using equation labels, make a sequence of the two equations resulting from differentiation, and press the Enter key.

•	Context Panel: Solve≻Solve for Variables≻ $u_{x}, z_{x}$ Enter as per Figure 4.3.25(a).

Figure 4.3.25(a) Variables dialog

$,$

$2 x + 4 {z (x, y)}^{3} (\frac{\partial}{\partial x} z (x, y)) + 5 {u (x, y, z)}^{4} (\frac{\partial}{\partial x} u (x, y, z)) = 0, 1 + 3 {z (x, y)}^{2} (\frac{\partial}{\partial x} z (x, y)) = 0$

(5)

$\overset{solve (specified)}{\to}$

$\{\frac{\partial}{\partial x} u (x, y, z) = \frac{2}{15} \frac{- 3 x + 2 z (x, y)}{{u (x, y, z)}^{4}}, \frac{\partial}{\partial x} z (x, y) = - \frac{1}{3 {z (x, y)}^{2}}\}$

(6)

Thus, $u_{x} = \frac{2}{15} \frac{2 z - 3 x}{u^{4}} = \frac{4 z - 6 x}{15 u^{4}}$ .

Obtain $u_{y}$ from first principles

•	Write the first equation with the appropriate dependencies made explicit.

•	Context Panel: Differentiate≻With Respect To≻ $y$

$x^{2} + y^{3} + z {(x, y)}^{4} + u {(x, y, z)}^{5} = 1$

$x^{2} + y^{3} + {z (x, y)}^{4} + {u (x, y, z)}^{5} = 1$

(7)

$\overset{differentiate w.r.t. y}{\to}$

$3 y^{2} + 4 {z (x, y)}^{3} (\frac{\partial}{\partial y} z (x, y)) + 5 {u (x, y, z)}^{4} (\frac{\partial}{\partial y} u (x, y, z)) = 0$

(8)

•	Write the second equation with the appropriate dependencies made explicit.

•	Context Panel: Differentiate≻With Respect To≻ $y$

$x + y^{2} + z {(x, y)}^{3} = 1$

$x + y^{2} + {z (x, y)}^{3} = 1$

(9)

$\overset{differentiate w.r.t. y}{\to}$

$2 y + 3 {z (x, y)}^{2} (\frac{\partial}{\partial y} z (x, y)) = 0$

(10)

•	Using equation labels, make a sequence of the two equations resulting from differentiation, and press the Enter key.

•	Context Panel: Solve≻Solve for Variables≻ $u_{y}, z_{y}$ Enter as per Figure 4.3.25(b).

Figure 4.3.25(b) Variables dialog

$,$

$3 y^{2} + 4 {z (x, y)}^{3} (\frac{\partial}{\partial y} z (x, y)) + 5 {u (x, y, z)}^{4} (\frac{\partial}{\partial y} u (x, y, z)) = 0, 2 y + 3 {z (x, y)}^{2} (\frac{\partial}{\partial y} z (x, y)) = 0$

(11)

$\overset{solve (specified)}{\to}$

$\{\frac{\partial}{\partial y} u (x, y, z) = \frac{1}{15} \frac{y (- 9 y + 8 z (x, y))}{{u (x, y, z)}^{4}}, \frac{\partial}{\partial y} z (x, y) = - \frac{2}{3} \frac{y}{{z (x, y)}^{2}}\}$

(12)

Thus, $u_{y} = \frac{1}{15} \frac{y (8 z - 9 y)}{u^{4}} = \frac{8 y z - 9 y^{2}}{15 u^{4}}$ .

Maple Solution - Coded

Apply the implicitdiff command

$implicitdiff (\{x^{2} + y^{3} + z^{4} + u^{5} = 1, x + y^{2} + z^{3} = 1\}, \{u, z\}, \{u, z\}, x)$

$\{D_{1} (u) = \frac{2}{15} \frac{2 z - 3 x}{u^{4}}, D_{1} (z) = - \frac{1}{3 z^{2}}\}$

$implicitdiff (\{x^{2} + y^{3} + z^{4} + u^{5} = 1, x + y^{2} + z^{3} = 1\}, \{u, z\}, \{u, z\}, y)$

$\{D_{2} (u) = \frac{1}{15} \frac{y (8 z - 9 y)}{u^{4}}, D_{2} (z) = - \frac{2}{3} \frac{y}{z^{2}}\}$

From the first calculation, obtain $u_{x} = \frac{2}{15} \frac{2 z - 3 x}{u^{4}}$ ; and from the second, $u_{y} = \frac{1}{15} \frac{y (8 z - 9 y)}{u^{4}}$ .

A solution from first principles can also be constructed with the implicitdiff command.

Apply the implicitdiff command to each of the two given equations to obtain $u_{x}$

$implicitdiff (\{x^{2} + y^{3} + z^{4} + u^{5} = 1\}, \{u, z\}, \{u, z\}, x)$

$\{D_{1} (u) = - \frac{4}{5} \frac{z^{3} D_{1} (z)}{u^{4}} - \frac{2}{5} \frac{x}{u^{4}}, D_{1} (z) = D_{1} (z)\}$

$implicitdiff (\{x + y^{2} + z^{3} = 1\}, \{z\}, \{z\}, x)$

$\{D_{1} (z) = - \frac{1}{3 z^{2}}\}$

Use the eval command to substitute $z_{x}$ into the expression for $u_{x}$

$eval (,)$

$\{- \frac{1}{3 z^{2}} = - \frac{1}{3 z^{2}}, D_{1} (u) = \frac{4}{15} \frac{z}{u^{4}} - \frac{2}{5} \frac{x}{u^{4}}\}$

From this, extract $u_{x} = \frac{4}{15} \frac{z}{u^{4}} - \frac{2}{5} \frac{x}{u^{4}} = \frac{2}{15} \frac{2 z - 3 x}{u^{4}}$ .

Apply the implicitdiff command to each of the two given equations to obtain $u_{y}$

$implicitdiff (\{x^{2} + y^{3} + z^{4} + u^{5} = 1\}, \{u, z\}, \{u, z\}, y)$

$\{D_{2} (u) = - \frac{4}{5} \frac{z^{3} D_{2} (z)}{u^{4}} - \frac{3}{5} \frac{y^{2}}{u^{4}}, D_{2} (z) = D_{2} (z)\}$

$implicitdiff (\{x + y^{2} + z^{3} = 1\}, \{z\}, \{z\}, y)$

$\{D_{2} (z) = - \frac{2}{3} \frac{y}{z^{2}}\}$

Use the eval command to substitute $z_{y}$ into the expression for $u_{y}$

$eval (,)$

$\{- \frac{2}{3} \frac{y}{z^{2}} = - \frac{2}{3} \frac{y}{z^{2}}, D_{2} (u) = \frac{8}{15} \frac{y z}{u^{4}} - \frac{3}{5} \frac{y^{2}}{u^{4}}\}$

From this, extract $u_{y} = \frac{8}{15} \frac{y z}{u^{4}} - \frac{3}{5} \frac{y^{2}}{u^{4}} = \frac{1}{15} \frac{y (8 z - 9 y)}{u^{4}}$ .

<< Previous Example Section 4.3 Next Example >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com

Download Help Document

Maple

Maple Add-Ons

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Maple

Powerful math software that is easy to use

Maple Add-Ons

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim