Subjects

Calculus 1

Integration and Accumulation of Change

Applying Properties of Definite Integrals

•

Nov 23, 2025

•

2 pages

Master Implicit Differentiation for Tangent Lines

Michelle Baxcajay

@ichelleaxcajay_chaa

Implicit differentiation expands our calculus toolkit beyond explicit functions, allowing... Show more

# UNIT 3 - Differentiation: Composite, Implicit, and Inverse Functions
Lesson 2-Implicit differentiation

Explicit equation
y= x² +2

Implic

Implicit Differentiation Basics

When dealing with equations where y isn't isolated $like x²+y²=1$ , we need implicit differentiation. Unlike explicit equations $y=x²+2$ , implicit equations require special handling when taking derivatives.

The key difference appears when differentiating terms containing y. For these terms, we must apply the chain rule and multiply by dy/dx. For example, when differentiating y², we get 2y· $dy/dx$ instead of just 2y.

To find dy/dx using implicit differentiation:

Differentiate both sides of the equation with respect to x
Add dy/dx whenever differentiating a term with y
Gather all dy/dx terms on one side
Solve for dy/dx by isolating it

💡 Think of dy/dx as a variable you're solving for. When you see y in the equation, remember that y depends on x, so you need the chain rule and must include dy/dx in your derivative.

For example, to find dy/dx for y² - 5x³ = 3y:

Differentiate: 2y $dy/dx$ - 15x² = 3 $dy/dx$
Gather dy/dx terms: 2y $dy/dx$ - 3 $dy/dx$ = 15x²
Factor out dy/dx: $2y - 3$ $dy/dx$ = 15x²
Solve: dy/dx = 15x²/ $2y - 3$

Applications of Implicit Differentiation

Implicit differentiation helps us find tangent lines to curves that aren't functions. For instance, with a circle x² + y² = 4, we can find the slope at any point without rewriting the equation.

To find tangent lines, first differentiate implicitly to get the formula for dy/dx. For the circle equation, we get 2x + 2y $dy/dx$ = 0, which simplifies to dy/dx = -x/y. At the point (1,√3), the slope would be -1/√3.

Horizontal and vertical tangent lines are special cases in implicit differentiation:

Horizontal tangent lines occur when dy/dx = 0
Vertical tangent lines occur when dy/dx is undefined $denominator = 0$

🔑 Implicit differentiation extends your ability to analyze curves that aren't functions. This technique is essential for understanding complex shapes and relationships in advanced calculus.

When solving implicit differentiation problems, maintain a systematic approach:

Differentiate each term carefully
Track where dy/dx appears
Solve algebraically to isolate dy/dx
Substitute specific points if needed to find numerical slopes

This technique works for a wide variety of equations including trigonometric relationships $sin(x+y)=2x$ , logarithmic equations $ln(y³)=5x+3$ , and exponential forms $5x²-e⁴ʸ=-6$ .

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Subjects

Calculus 1

Integration and Accumulation of Change

Applying Properties of Definite Integrals

Calculus 1

•

Nov 23, 2025

•

2 pages

Master Implicit Differentiation for Tangent Lines

Michelle Baxcajay

@ichelleaxcajay_chaa

Implicit differentiation expands our calculus toolkit beyond explicit functions, allowing us to find derivatives when y isn't isolated. This technique is crucial for handling complex equations where solving for y would be difficult or impossible, such as circles, ellipses, and... Show more

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Implicit Differentiation Basics

To find dy/dx using implicit differentiation:

Differentiate both sides of the equation with respect to x
Add dy/dx whenever differentiating a term with y
Gather all dy/dx terms on one side
Solve for dy/dx by isolating it

💡 Think of dy/dx as a variable you're solving for. When you see y in the equation, remember that y depends on x, so you need the chain rule and must include dy/dx in your derivative.

For example, to find dy/dx for y² - 5x³ = 3y:

Differentiate: 2y $dy/dx$ - 15x² = 3 $dy/dx$
Gather dy/dx terms: 2y $dy/dx$ - 3 $dy/dx$ = 15x²
Factor out dy/dx: $2y - 3$ $dy/dx$ = 15x²
Solve: dy/dx = 15x²/ $2y - 3$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Applications of Implicit Differentiation

Implicit differentiation helps us find tangent lines to curves that aren't functions. For instance, with a circle x² + y² = 4, we can find the slope at any point without rewriting the equation.

Horizontal and vertical tangent lines are special cases in implicit differentiation:

Horizontal tangent lines occur when dy/dx = 0
Vertical tangent lines occur when dy/dx is undefined $denominator = 0$

🔑 Implicit differentiation extends your ability to analyze curves that aren't functions. This technique is essential for understanding complex shapes and relationships in advanced calculus.

When solving implicit differentiation problems, maintain a systematic approach:

Differentiate each term carefully
Track where dy/dx appears
Solve algebraically to isolate dy/dx
Substitute specific points if needed to find numerical slopes

This technique works for a wide variety of equations including trigonometric relationships $sin(x+y)=2x$ , logarithmic equations $ln(y³)=5x+3$ , and exponential forms $5x²-e⁴ʸ=-6$ .

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What is the Knowunity AI companion?

Where can I download the Knowunity app?

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Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

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4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user