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Section The Chain Rule
Worksheet DC-2 Reading
Objectives
Things you should understand and know:
Things you should be able to do:
Calculate the derivative of a function with one independent variable using the Chain Rule
Calculate the derivative of a function with two independent variables using the Chain Rule
Use implicit differentiation on a function of two or more variables
You will be assessed on the following criteria:
Problem structure: Solution should be organized and easy to follow, with clear, complete steps that follow progressively from one to the next.
Chain Rule: Understand the concept of the Chain Rule and its applications. Calculate the derivative of a function with one or two independent variables using the Chain Rule.
Implicit Differentiation: Use implicit differentiation on a function of two or more variables.
All of the readings for this course come from the OpenStax Calculus Volume 3 textbook, which is available for free online at
https://openstax.org/details/books/calculus-volume-3 . You can read the textbook online or download a PDF version. The textbook is also available in print from various retailers if you prefer a physical copy.
For this lesson, you should read sections
4.5 The Chain Rule . You should read through the sections carefully, making sure to understand the definitions, examples, and key ideas. You should also work through the exercises at the end of each section to test your understanding and practice applying the concepts. Some of the exercises at the end of the section are assigned in the WP assignments.
Worksheet DC-2 Written Practice
INFORMATION ABOUT ALL WRITTEN PRACTICE (WP) ASSIGNMENTS:
THE WP ASSIGNMENTS ARE DESIGNED TO PREPARE YOU FOR THE QUIZZES AND ASSESSMENTS
Work on every problem on every assignment.
If you get stuck or do not understand a solution, ask questions.
For each problem, try it once or twice before looking at solutions or asking for help.
Mark the problems you did correctly on the first try.
Highlight problems where you got help (using solutions, videos, a tutor, etc) -These are the problem types that you will need more practice on to be ready for the assessments.
Understand that just copying the solutions instead of working through the problems will greatly reduce your chances of success in this class.
The Chain Rule: Complete problems 217, 223, 243, 245, 251, 257
Worksheet DC-2 Sample Outcomes
For the quiz on this outcome you would be given 40 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
1. Given
\(u(x,y)=x^4-3xy+1\) and
\(x=2t,y=t^3\)
(a) Find
\(\frac{du}{dt}\text{.}\)
Hint .
Use the chain rule:
\(\frac{du}{dt} = \frac{\partial u}{\partial x}\frac{dx}{dt} + \frac{\partial u}{\partial y}\frac{dy}{dt}\text{.}\)
Answer .
\(\frac{du}{dt} = 4x^3\frac{dx}{dt} - 3y\frac{dx}{dt} - 3x\frac{dy}{dt}\)
Solution .
First, compute the partial derivatives of
\(u\text{:}\)
\(\frac{\partial u}{\partial x} = 4x^3 - 3y\)
\(\frac{\partial u}{\partial y} = -3x\)
Next, compute the derivatives of
\(x\) and
\(y\) with respect to
\(t\text{:}\)
\(\frac{du}{dt} = (4x^3 - 3y)(2) + (-3x)(3t^2)\)
Put everything in terms of
\(t\text{:}\)
\(\frac{du}{dt} = 8x^3 - 6y - 9xt^2\)
\(\frac{du}{dt} = 8(2t)^3 - 6(t^3) - 9(2t)(t^2)\)
\(\frac{du}{dt} = 64t^3 - 6t^3 - 18t^3\)
\(\frac{du}{dt} = 40t^3\)
(b) Find
\(\frac{d^2u}{dt^2}\text{.}\)
Hint .
First find
\(\frac{du}{dt}\text{,}\) then differentiate it with respect to
\(t\text{.}\)
Answer .
\(\frac{d^2u}{dt^2} = 120t^2\)
Solution .
From the previous calculation, we have:
\(\frac{du}{dt} = 40t^3\)
Differentiate with respect to
\(t\text{:}\)
\(\frac{d^2u}{dt^2} = 120t^2\)
2. Given
\(x^2+y^2+z^2=3xyz\)
(a) Find
\(\frac{\partial z}{\partial x}\text{.}\)
Hint .
Use implicit differentiation (this is one method of solving).
Answer .
\(\frac{\partial z}{\partial x} = \frac{3yz - 2x}{2z - 3xy}\)
Solution .
Implicitly differentiate the equation with respect to
\(x\text{:}\)
\(2x + 2z\frac{\partial z}{\partial x} = 3yz + 3xy\frac{\partial z}{\partial x}\)
Solve for
\(\frac{\partial z}{\partial x}\text{:}\)
\(2z\frac{\partial z}{\partial x} - 3xy\frac{\partial z}{\partial x} = 3yz - 2x\)
\(\frac{\partial z}{\partial x}(2z - 3xy) = 3yz - 2x\)
\(\frac{\partial z}{\partial x} = \frac{3yz - 2x}{2z - 3xy}\)
(b) Find
\(\frac{\partial z}{\partial y}\text{.}\)
Hint .
Here we can rewrite the equation as
\(f(x,y,z) = x^2 + y^2 + z^2 - 3xyz = 0\) and use the chain rule.
Answer .
\(\frac{\partial z}{\partial y} = \frac{3xz - 2y}{2z - 3xy}\)
Solution .
We first rewrite the equation as
\(f(x,y,z) = x^2 + y^2 + z^2 - 3xyz = 0\text{.}\)
Then, we find the partial derivatives
\(\frac{\partial f}{\partial y}\) and
\(\frac{\partial f}{\partial z}\text{:}\)
\(\frac{\partial f}{\partial y} = 2y - 3xz\)
\(\frac{\partial f}{\partial z} = 2z - 3xy\)
Substitute into the chain rule formula:
\(\frac{\partial z}{\partial y} = -\frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial z}}\)
\(\frac{\partial z}{\partial y} = -\frac{2y - 3xz}{2z - 3xy}\)
\(\frac{\partial z}{\partial y} = \frac{3xz - 2y}{2z - 3xy}\)
