Back To Course

Math 104: Calculus14 chapters | 115 lessons | 11 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Robert Egan*

How do you define the rate of change when your function has variables that cannot be separated? Learn how implicit differentiation can be used to find dy/dx even when you don't have y=f(x)!

I have an Uncle Joe who's a farmer. He really likes math, so he told me about his plot of land. He said his land extends *x* meters to the east and *y* meters to the north. He said, 'You know, the area of that land I have is (*x*)(*y*).' That's because his land is just a rectangle, and the border, or perimeter, around his land is just 2*x* + 2*y*; we've got *x* + *y* + *x* + *y*. Now Uncle Joe told me that his land always satisfies one condition. That is that the area of his land is always equal to half of the perimeter. In other words, (*x*)(*y*) = 1/2(2*x* + 2*y*) = *x* + *y*. That's great, Uncle Joe! You love math - what do you need me for?

Well, Uncle Joe always wants this equation, (*x*)(*y*) = *x* + *y*, to be true. He's in the land business. He wants to know if he buys more land to the east - so if he changes *x* - how much does he have to change *y* to keep this equation true? He wants to know *dy/dx* - how much *y* should change while *x* is changing. Oh, so Uncle Joe wants me to calculate a derivative. I can do this.

Okay, find *dy/dx* for (*x*)(*y*) = *x* + *y*. The first thing I want to do is set up *y*=*f(x)* ... uh-oh, I can't do that; I can't separate *x* and *y* to different sides of this particular equation. This is going to make Farmer Joe really unhappy. Well, maybe there's another way to find *dy/dx*. I remember that to find *dy/dx* of *f(x)*, I wrote *y*=*f(x)* and differentiated both sides. I got *dy/dx* = *d/dx f(x)*. Why don't I try that here?

*d/dx*(*xy*) = *d/dx*(*x* + *y*). For *d/dx*(*xy*), that looks like something I need to use the product rule on: *d/dx*(*x*)*y* + *d/dx*(*y*)*x*. The first term is 1 * *y*, or *y*, plus *dy/dx*(*x*). I can write *dy/dx* as *y`*, then the whole left-hand side of my equation becomes *y* + *xy`*. The right-hand side of my equation is *d/dx*(*x* + *y*). I can split that up and write *d/dx*(*x*) + *d/dx*(*y*). That's just 1 + *dy/dx* = 1 + *y`*. My entire equation - when I differentiate (*x*)(*y*) = *x* + *y* - is *y* + *xy`* = 1 + *y`*. Alright, but I'm trying to find *dy/dx*, or *y`*.

So let's solve this for *y`*. First, let's collect all the terms, move all the *y`* terms to the left-hand side, *xy`* - *y`* = 1 - *y*. Let's factor out *y`*, so I have *y`*(*x* - 1) = 1 - *y*. Then let's divide the left- and right-hand sides by (*x* - 1), and we end up with *y`* = (1 - *y*) / (*x* - 1). Whew, that was a lot of work for Uncle Joe!

What we just did is an example of **implicit differentiation**. For implicit differentiation, we can follow these steps:

- Differentiate both sides.
- Collect
*y`*terms to one side of the equation. - Factor
*y`*out of the terms. - Solve for
*y`*.

Let's do an example. Let's say we have *y* = *ye*^*x* + *x*, and let's say we're trying to find *y`*, or *dy/dx*.

Our first step is to differentiate both sides. *d/dx*(*y*) = *d/dx*(*ye*^*x* + *x*). So *d/dx*(*y*) is just *dy/dx*. *d/dx* of the right-hand side is a little bit more complicated. Let's look at this first term, *ye*^*x*. The derivative of *ye*^*x* is *y*(*d/dx*(*e*^*x*)) + *e*^*x*(*d/dx*(*y*)). That's just using the product rule. This first term becomes *ye*^*x* + *e*^*x* *dy/dx*. The second term is just *x*, and if we take the derivative of *x* with respect to *x*, we get 1. Alright, so let's write *dy/dx* as *y`*: *y`* = *ye*^*x* + *e*^(*x*)*y`* + 1. Great, we've differentiated both sides.

Now, let's collect all of the *y`* terms on one side of the equation: *y`* - *e*^(*x*)*y`* = *ye*^*x* + 1. So all I've done is move *e*^(*x*)*y`* to the left-hand side. Fantastic, halfway there. Now I'm going to factor out *y`*: *y`*(1 - *e*^*x*) = *ye*^*x* + 1. Finally, I'm going to solve for *y`* by dividing everything by (1 - *e*^*x*), so *y`* = *ye*^*x* + 1 / (1 - *e*^*x*). This is great, as long as *x* doesn't equal zero. If *x*=0, we're trying to divide by zero, and we can't do that.

This is how we find *y`*, or *dy/dx*, for another case where we have an implicit equation, *y* = *ye*^*x* + *x*.

**Implicit differentiation** is what you use when you have *x* and *y* on both sides of an equation and you're looking for *dy/dx*. We did this in the case of Farmer Joe's land when he gave us the equation (*x*)(*y*) = *x* + *y*.

To do implicit differentiation, we:

- Differentiate both sides.
- Collect
*y`*terms to one side of the equation. - Factor
*y`*out of those terms. - Solve for
*y`*.

We end up with *y`* as some function of *x* and *y*.

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
11 in chapter 7 of the course:

Back To Course

Math 104: Calculus14 chapters | 115 lessons | 11 flashcard sets

- Go to Continuity

- Go to Limits

- Using Limits to Calculate the Derivative 8:11
- The Linear Properties of a Derivative 8:31
- Calculating Derivatives of Trigonometric Functions 7:20
- Calculating Derivatives of Polynomial Equations 10:25
- Calculating Derivatives of Exponential Equations 8:56
- Using the Chain Rule to Differentiate Complex Functions 9:40
- Differentiating Factored Polynomials: Product Rule and Expansion 6:44
- When to Use the Quotient Rule for Differentiation 7:54
- Understanding Higher Order Derivatives Using Graphs 7:25
- Calculating Higher Order Derivatives 9:24
- How to Find Derivatives of Implicit Functions 9:23
- Applying the Rules of Differentiation to Calculate Derivatives 11:09
- Optimization Problems in Calculus: Examples & Explanation 10:45
- Go to Calculating Derivatives and Derivative Rules

- English 310: Short Stories
- Nurse Entrance Test (NET): Exam Prep & Study Guide
- CCMA Basic Exam: Study Guide & Test Prep
- Personalized Learning in the Classroom
- Certified Emergency Nurse (CEN): Study Guide & Exam Prep
- 19th-Century British Short Stories
- 19th & Early 20th-Century American Naturalist Short Stories
- 19th-Century Russian Realism in Short Stories
- Early 20th-Century Feminist Short Stories
- Medical Records & HIPAA
- Professional Publications in Literacy
- Dyslexia Programs in Texas
- Study.com's Teacher Edition
- Study.com School Plans
- Study.com's Virtual Classrooms
- How to Set Up a Class and Invite Students in Your Study.com Virtual Classroom
- How to View Grades and Export CSVs in Your Study.com Virtual Classroom

- Tree Diagrams in Math: Definition & Examples
- How to Square a Trinomial
- How to Find the Least Common Multiple of Expressions
- Converting 1 Radian to Degrees
- Rip Van Winkle Literary Criticism
- Working Papers in the Audit Process: Definition & Development
- German Genitive Pronouns
- Treaty of Paris Lesson Plan
- Quiz & Worksheet - The Advancement of Learning by Francis Bacon
- Quiz & Worksheet - Practice Graphing Radical Functions
- Quiz & Worksheet - Finding the Major Axis of an Ellipse
- Quiz & Worksheet - Types of Triangles & Their Properties
- Quiz & Worksheet - Point of Care Technology in Healthcare
- Developing Presentation Skills Flashcards
- Hypothesis Testing in Statistics Flashcards

- Religion 101: Intro to World Religions
- High School Biology: Tutoring Solution
- Glencoe Algebra 1: Online Textbook Help
- NY Regents Exam - Global History and Geography: Tutoring Solution
- Introduction to Business Law: Certificate Program
- Strategies for Economic Analysis
- NES ELA: Comma Rules
- Quiz & Worksheet - Rude Mechanicals in A Midsummer Night's Dream
- Quiz & Worksheet - Secondary Sex Characteristics
- Quiz & Worksheet - Exclamatory Sentence
- Quiz & Worksheet - Emulsion
- Quiz & Worksheet - Understanding There Will Come Soft Rains

- Slant Rhyme in Poetry: Definition & Examples
- Construction Project Management: Definition & Examples
- Opportunity Cost Lesson Plan
- How to Register for the TEAS Test
- 8th Grade Colorado Science Standards
- Questions to Ask Your College Advisor
- Child Advocacy Groups
- Praxis Tests in Arkansas
- Florida State Standards for Math
- How to Be Successful in College: Avoid Debt & Remedial Courses
- How to Create Assignments in Your Study.com Virtual Classroom
- States of Matter Lesson Plan

Browse by subject