#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 $y = x^2$, where $y$ is isolated. But what about equations like $xy^2 = xy + 1$? That's where implicit differentiation comes in! It's a method to find derivatives when $y$ isn't explicitly defined in terms of $x$. Think of it as finding the slope of a curve even when the equation is a bit tangled. 🔄

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1. Notate: Indicate that you're differentiating both sides with respect to $x$. $\frac{d}{dx} (\text{equation}) = \frac{d}{dx} (\text{other side})$
2. Differentiate: Apply derivative rules (power rule, product rule, chain rule). Remember, $\frac{dx}{dx} = 1$, but $\frac{dy}{dx}$ remains as $\frac{dy}{dx}$ or $y'$.
3. Isolate: Solve for $\frac{dy}{dx}$. You'll often need to factor out $\frac{dy}{dx}$.

#Example: The Unit Circle ⭕

Let's find $\frac{dy}{dx}$ for $x^2 + y^2 = 1$:

1. Notate:

latex
$$
\frac{d}{dx}(x^2 + y^2) = \f...