1. Differentiate Both Sides: Apply $\frac{d}{dx}$ to every term.
2. The Y-Witness: Whenever you differentiate $y$, multiply by $\frac{dy}{dx}$.
3. Group the Primes: Move every term containing $\frac{dy}{dx}$ to the left.
4. Isolate: Factor out the $\frac{dy}{dx}$ and solve the equation.
Find $\frac{dy}{dx}$ for each equation.
$x^2 + y^2 = 36$
$y^3 = x + 10$
$x^2 + 5y = 100$
$x^2 - y^2 = 1$ (A Hyperbola)
$y^2 + y = x^2$
In Part I, what is the derivative of the constant "36"? Why does it become 0? Does the size of the constant affect the Speed of the relationship?
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Find $\frac{dy}{dx}$ for $xy = 10$.
Find the slope of the ellipse $x^2 + 4y^2 = 100$.
1. Differentiate...
2. Group...
3. Solve...
Objective: Explain "Implicit" to a younger student using a knot.
The Activity:
1. Tie two strings together in a messy knot.
2. Pull one string ($x$).
3. Ask: "Did I pull the other string ($y$)? No. But did it move? Yes!"
The Lesson: "In life, things are often knotted together. Even if we can't see the connection, pulling one part moves the other. Math can see the connection even inside the knot."
Response: ___________________________________________________________