Implicit Differentiation

Benjamin Wright
7 min read
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Study Guide Overview
This guide covers implicit differentiation, focusing on how to find when y is not explicitly defined in terms of x. It outlines the steps for implicit differentiation, including using the chain rule and product rule. The guide provides examples, practice problems involving tangent lines, and common mistakes to avoid. Finally, it offers exam tips and strategies, highlighting high-priority topics like combining implicit differentiation with related rates problems.
AP Calculus AB/BC: Implicit Differentiation - Your Ultimate Guide 🚀
Hey there, future calculus masters! 👋 Let's dive into implicit differentiation, a key technique that'll help you ace those tricky problems. This guide is designed to be your go-to resource, especially the night before the exam. Let's make sure you're feeling confident and ready!
Implicit Differentiation: Unlocking Hidden Derivatives
What is Implicit Differentiation? 🤔
We're used to explicit equations like , where is isolated. But what about equations like ? That's where implicit differentiation comes in! It's a method to find derivatives when isn't explicitly defined in terms of . Think of it as finding the slope of a curve even when the equation is a bit tangled. 🔄
Key Idea: Differentiate both sides of the equation with respect to , remembering to use the chain rule for both and . Then, solve for .
Chain Rule Reminder: When differentiating a term involving , remember to multiply by because is a function of !
Steps for Implicit Differentiation
- Notate: Indicate that you're differentiating both sides with respect to .
- Differentiate: Apply derivative rules (power rule, product rule, chain rule). Remember, , but remains as or .
- Isolate: Solve for . You'll often need to factor out .
Example: The Unit Circle ⭕
Let's find for :
- Notate:
<math-block>\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(1)</math-block>
- Differentiate:
<math-block>2x\frac{dx}{dx} + 2y\frac{dy}{dx} = 0</math-block>
<math-block>2x + 2y\frac{dy}{dx} = 0</math-block>
- Isolate:
<math-block>2y\frac{dy}{dx} = -2x</math-block>
<math-block>\frac{dy}{dx} = \frac{-2x}{2y} = \frac{-x}{y}</math-block>
Quick Check: The derivative gives the slope of the unit circle at any point .
For instance, at the point , , and at the point , .
Graph of
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✍️ Implicit Differentiation Practice Problems
Let's solidify your understanding with some practice problems. These are designed to mimic what you'll see on the AP exam.
1) Implicit Derivatives and Tangent Lines
Consider the curve given by .
(a) Calculate .
(b) Write an equation for the line tangent to the curve at the point .
a) Calculate the derivative
- Notate:
<math-block>\frac{d}{dx}(y^3 - xy) = \frac{d}{dx}(2)</math-block>
- Differentiate:
<math-block>3y^2\frac{dy}{dx} - (x\frac{dy}{dx} + y) = 0</math-block>
- Isolate:
<math-block>3y^2\frac{dy}{dx} - x\frac{dy}{dx} - y = 0</math-block>
<math-block>\frac{dy}{dx}(3y^2 - x) = y</math-block>
<math-block>\frac{dy}{dx} = \frac{y}{3y^2 - x}</math-block>
b) Equation for the tangent line
Tangent Line Equation: Remember the point-slope form: , where is the slope (derivative) at the point .
We have , . Now find the slope at :
<math-block>\frac{dy}{dx} = \frac{1}{3(1)^2 - (-1)} = \frac{1}{4}</math-block>
The tangent line equation is:
<math-block>y - 1 = \frac{1}{4}(x + 1)</math-block>
Common Mistake: Forgetting to use the product rule when differentiating terms like or forgetting when differentiating y. Always double-check your steps!
Product Rule Reminder: . Don't forget it when you have two functions multiplied together!
You aced it! 3/3 points on the FRQ. 🌟
🎉 Closing Thoughts
You've now got the hang of implicit differentiation! It's a powerful tool that will come up in many different forms on the exam. Keep practicing, and you'll be ready for anything. Remember, you've got this! 💪
Encouraging GIF with animated ice cream
Image Courtesy of Giphy
Final Exam Focus 🎯
-
High-Priority Topics: Implicit differentiation is frequently combined with related rates and tangent line problems.
-
Common Question Types: Expect to find , use it to find tangent lines, and solve related rates problems.
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Time Management: Practice solving problems quickly. Focus on identifying the key steps and avoid getting bogged down in algebra.
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Common Pitfalls: Watch out for product rule and chain rule errors. Always double-check your work.
Exam Strategy: If you get stuck, write down what you know and try to apply the steps of implicit differentiation. Partial credit is often given for correct setup and differentiation.
Practice Question
Practice Questions
Multiple Choice Questions
-
If , then at the point (3, 4) is: (A) -3/4 (B) 3/4 (C) -4/3 (D) 4/3
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If , then at the point (2, 1) is: (A) -1/4 (B) -1/3 (C) -1/2 (D) 1/2
Free Response Question
Consider the curve defined by .
(a) Find .
(b) Write an equation for the line tangent to the curve at the point (3, 3).
(c) Find all points on the curve where the tangent line is horizontal.
FRQ Scoring Breakdown
(a) Finding the derivative (5 points):
* 1 point: Correctly applying the power rule and chain rule on and
* 2 points: Correctly applying the product rule on 6xy
* 1 point: Correctly isolating
* 1 point: Correctly simplifying the final answer
(b) Tangent Line (2 points): * 1 point: Correctly evaluating the derivative at (3,3) * 1 point: Correctly writing the equation of the tangent line
(c) Horizontal Tangents (2 points): * 1 point: Setting equal to 0 * 1 point: Correctly finding all points
Answers
MCQ
- (A)
- (B)
FRQ
(a)
(b)
(c) (0,0) and (4, 2)

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