The Chain Rule and Implicit Differentiation

Section 12.1 The Chain Rule and Implicit Differentiation

In this section, we extend the chain rule to functions of several variables. We also revisit implicit differentiation, and extend it to situations involving functions of several variables.

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Subsection 12.1.1 Review of the Chain Rule for Functions of a Single Variable

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Recall the chain rule for composite functions: if

$y = f(x)$ and

$x = g(t)$ are both differentiable functions, then

$y$ is a differentiable function of

$t\text{,}$ and

$\frac{dy}{dt} = \frac{dy}{dx} \frac{dx}{dt}.$

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Subsection 12.1.2 The Chain Rule for Functions of Several Variables

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Functions of Two Variables:

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Suppose that

$z = f(x,y)$ is a differentiable function of

$x$ and

$y\text{,}$ where

$x=g(t)$ and

$y=h(t)$ are both differentiable functions of

$t\text{.}$ Then

$z$ is a differentiable function of

$t\text{,}$ and

$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}.$

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Functions of Three Variables:

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Suppose that

$w = f(x,y,z)$ is a differentiable function of

$x\text{,}$

$y\text{,}$ and

$z\text{,}$ and

$x\text{,}$

$y\text{,}$ and

$z$ are differentiable functions of

$t\text{.}$ Then

$w$ is a differentiable function of

$t\text{,}$ and

$\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} + \frac{\partial w}{\partial z} \frac{dz}{dt}.$

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The chain rule extends naturally for functions of more than three variables.

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In the following video, we work through some examples.

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Figure 12.1.1. Video demonstrating the chain rule for functions of several variables.

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Subsection 12.1.3 Review of Implicit Differentiation

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Consider the folium of Descartes, the curve defined implicitly by the equation

$x^3 + y^3 = 6xy,$ shown below.

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Figure The folium of Descartes, given implicitly by $x^3 + y^3 = 6xy\text{.}$ Source: Biocalculus: Calculus for the Life Sciences, James Stewart and Troy Day, Cengage Learning, 2015.

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It is not possible to isolate

$y$ and write it in terms of

$x\text{.}$ To determine

$\displaystyle \frac{dy}{dx}\text{,}$ we can differentiate both sides of the equation with respect to

$x\text{,}$ keeping in mind that

$y$ is a function of

$x\text{:}$

$\frac{d}{dx}(x^3 + y^3) = \frac{d}{dx} (6xy).$ This gives

$3x^2 + 3y^2 \frac{dy}{dx} = 6x \frac{dy}{dx} + 6y.$ Re-arranging and solving for

$\displaystyle \frac{dy}{dx}$ (left as an exercise for the reader) gives

$\frac{dy}{dx} = \frac{2y-x^2}{y^2-2x}.$

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Subsection 12.1.4 Implicit Differentiation Done Differently

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Another way to find

$\displaystyle \frac{dy}{dx}$ for the folium of Descartes,

$x^3 + y^3 = 6xy\text{,}$ is to define the function

$F(x,y) = x^3 + y^3 - 6xy.$ The folium of Descartes then is given by

$F(x,y) = 0.$

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Using the chain rule from above, and differentiating both sides with respect to

$x\text{,}$ we have

$\frac{\partial F}{\partial x} \frac{dx}{dx} + \frac{\partial F}{\partial y} \frac{dy}{dx} = 0.$

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But

$\displaystyle \frac{dx}{dx} = 1\text{.}$ Provided

$\displaystyle \frac{\partial F}{\partial y} \ne 0\text{,}$ we can isolate

$\displaystyle \frac{dy}{dx}$ and get

$\frac{dy}{dx} = - \frac{ \frac{\partial F}{\partial x} }{ \frac{\partial F}{\partial y} } = - \frac{F_x}{F_y}.$

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Applying this result to the folium of Descartes, given by

$F(x,y) = x^3 + y^3 - 6xy\text{,}$ we have

$\begin{alignat*}{1} F_x &= 3x^2 - 6y, \\ F_y &= 3y^2 - 6x, \end{alignat*}$

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so that

$\frac{dy}{dx} = -\frac{F_x}{F_y} = - \frac{3x^2 - 6y}{3y^2 - 6x} = \frac{2y-3x^2}{y^2-2x},$ as before.

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Subsection 12.1.5 Implicit Differentiation Generalized

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Consider a surface

$z=f(x,y)$ defined implicitly by an equation of the form

$F(x,y,z) = 0.$

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Using the chain rule from above, and differentiating both sides with respect to

$x$ gives

$\frac{\partial F}{\partial x} \frac{dx}{dx} + \frac{\partial F}{\partial y} \frac{dy}{dx} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial x} = 0.$ But

$\displaystyle \frac{dx}{dx} = 1$ and

$\displaystyle \frac{dy}{dx} = 0$ (this is because both

$y$ and

$x$ are independent variables;

$y$ does not depend on

$x$ ). We thus have

$\frac{\partial F}{\partial x} + \frac{\partial F}{\partial z} \frac{\partial z}{\partial x} = 0.$ Provided

$\displaystyle \frac{\partial F}{\partial z} \ne 0\text{,}$ we can isolate

$\displaystyle \frac{\partial z}{\partial x}$ to get

$\frac{\partial z}{\partial x} = - \frac{ \frac{\partial F}{\partial x} }{ \frac{\partial F}{\partial z} } = - \frac{F_x}{F_z}.$

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Similarly (left as an exercise to the reader),

$\frac{\partial z}{\partial y} = - \frac{ \frac{\partial F}{\partial y} }{ \frac{\partial F}{\partial z} } = - \frac{F_y}{F_z}.$

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In the following video, we work through a specific example.

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Figure 12.1.3. Video demonstrating implicit differentiation for functions of several variables.

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Subsection 12.1.6 Summary

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The Chain Rule.

Suppose that $z = f(x,y)$ is a differentiable function of $x$ and $y\text{,}$ where $x=g(t)$ and $y=h(t)$ are both differentiable functions of $t\text{.}$ Then $z$ is a differentiable function of $t\text{,}$ and $\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}.$
Suppose that $w = f(x,y,z)$ is a differentiable function of $x\text{,}$ $y\text{,}$ and $z\text{,}$ and $x\text{,}$ $y\text{,}$ and $z$ are differentiable functions of $t\text{.}$ Then $w$ is a differentiable function of $t\text{,}$ and $\frac{dw}{dt} = \frac{\partial w}{\partial x} \frac{dx}{dt} + \frac{\partial w}{\partial y} \frac{dy}{dt} + \frac{\partial w}{\partial z} \frac{dz}{dt}.$

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Implicit Differentiation.

For a curve $y=f(x)$ defined implicitly by the equation $F(x,y) = 0\text{,}$ $\frac{dy}{dx} = - \frac{ \frac{\partial F}{\partial x} }{ \frac{\partial F}{\partial y} } = - \frac{F_x}{F_y}.$
For a surface $z=f(x,y)$ defined implicitly by the equation $F(x,y,z) = 0\text{,}$
$\begin{alignat*}{1} \frac{\partial z}{\partial x} &= - \frac{ \frac{\partial F}{\partial x} }{ \frac{\partial F}{\partial z} } = - \frac{F_x}{F_z}, \\ \frac{\partial z}{\partial y} &= - \frac{ \frac{\partial F}{\partial y} }{ \frac{\partial F}{\partial z} } = - \frac{F_y}{F_z}. \end{alignat*}$

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Subsection 12.1.7 Don't Forget

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Don't forget to return to eClass to complete the pre-class quiz.

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Subsection 12.1.8 Further Study

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Remember that the notes presented above only serve as an introduction to the topic. Further study of the topic will be required. This includes working through the pre-class quizzes, reviewing the lecture notes, and diligently working through the homework problems.

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As you study, you should reflect on the following learning outcomes, and critically assess where you are on the path to achieving these learning outcomes:

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Learning Outcomes

Apply the chain rule for functions of several variables.
When $F(x, y) = 0$ defines $y$ implicitly in terms of $x\text{,}$ determine the derivative of $y$ with respect to $x$ in terms of the partial derivatives of $F\text{.}$
When $F(x, y, z) = 0$ defines $z$ implicitly in terms of $x$ and $y\text{,}$ determine the partial derivative of $z$ with respect to $x$ and the partial derivative of $z$ with respect to $y$ in terms of the partial derivatives of $F\text{.}$

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The following references provide a good start for review and further study:

Learning Outcome	Video	Textbook Section
1	12.E1	9.4
2	12.E2	9.4
3	12.E2	9.4