In Problems 47-52, z is a function of x and y . Find (∂z)/(∂x) and (∂z)/(∂y) F(x, y, z)=x^2 z+y^

Name: Problem 48, Chapter 12: Calculus, Early Transcendentals
Uploaded: 2021-08-06T16:20:05-08:00
Duration: 1 min 9 s
Channel: Tyler Moulton
Description: In Problems 47-52, z is a function of x and y . Find (∂z)/(∂x) and (∂z)/(∂y) F(x, y, z)=x^2 z+y^2 z+x^3 y-10 z=0

step 1 In this problem, we are given that Z is a function of X and Y, and we want to find DZDX and DZDY

Question

In Problems $47-52, z$ is a function of $x$ and $y .$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ $$ F(x, y, z)=x^{2} z+y^{2} z+x^{3} y-10 z=0 $$

   In Problems $47-52, z$ is a function of $x$ and $y .$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$
$$ F(x, y, z)=x^{2} z+y^{2} z+x^{3} y-10 z=0 $$

Calculus, Early Transcendentals

Michael Sullivan,… 1st Edition

Chapter 12, Problem 48

Instant Answer

Solved by Expert Tyler Moulton

08/06/2021

Step 1

These are denoted as $F_x$, $F_y$, and $F_z$ respectively. Using the power rule of differentiation, we find that $$F_x = 2xz + y^2 + 3x^2y$$ $$F_y = 2yz + x^3$$ $$F_z = x^2 + y^2 - 10$$ Show more…

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In Problems $47-52, z$ is a function of $x$ and $y .$ Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ $$ F(x, y, z)=x^{2} z+y^{2} z+x^{3} y-10 z=0 $$

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Key Concepts

Implicit Differentiation

A technique used to find derivatives of functions defined by an equation in which the dependent variable is not isolated. This method involves differentiating both sides of the equation with respect to the independent variable while treating the dependent variable as an implicit function of that variable.

Partial Derivatives

These derivatives measure the rate at which a function of several variables changes with respect to one variable, while all other variables are held constant. They are fundamental in studying functions of multiple variables and are widely applied in optimization and modeling.

Chain Rule

A fundamental rule in calculus that is used to compute the derivative of composite functions. In the context of multivariable functions, it is essential when differentiating implicit functions, as it accounts for the indirect dependence of variables on one another.

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