Non-Functions: Curves like $x^2 + y^2 = r^2$ fail the Vertical Line Test. We cannot easily isolate for $y=f(x)$.
The Chain Rule Trick: We assume $y$ is some unknown function of $x$. So, taking the derivative of $y^2$ with respect to x requires multiplying by $\frac{dy}{dx}$ (Chain Rule!). $\frac{d}{dx}[y^2] = 2y \cdot \frac{dy}{dx}$.
Multiple Tangents: Because a single $x$ can have multiple $y$ values, the slope $\frac{dy}{dx}$ requires BOTH $x$ and $y$ coordinates to lock onto a single tangent line.
Tags
DerivativesImplicitGeometryChain Rule
DRAG ALONG THE CURVE TO EVALUATE THE TANGENT
Select Implicit Equation
Evaluation at (x, y) = (0.00, 5.00)
DERIVATIVE EQUATION
$\frac{dy}{dx} = -\frac{x}{y}$
INSTANTANEOUS SLOPE
Numerator0.00
Denominator5.00
$\frac{dy}{dx} = $0.00
Warning: Vertical Tangent! Because $\frac{dy}{dx} = -\frac{x}{y}$, anywhere the y-coordinate is $0$ (the x-intercepts), the slope is $\frac{x}{0}$, making it undefined (a perfectly vertical line).