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 π
Hey there, future calculus masters! π 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? π€
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. π
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 !
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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 β
Let's find for :
- Notate:
latex
$$
\frac{d}{dx}(x^2 + y^2) = \f...

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