The Knowledge Web

Explicit vs. Implicit

Until now, every equation has looked like $y = 3x^2 + 2x$ . This is Explicit. $y$ is explicitly defined in terms of $x$ .

But look at the equation of a circle: $x^2 + y^2 = 25$ . $y$ is tangled up with $x$ . This is Implicit.

The Orbital Inspector

x² + y² = 25

Click & Drag on Circle

Live Calculation

Because we differentiated implicitly, the slope formula $dy/dx$ requires both the $x$ and $y$ coordinates to work.

x coordinate3.54

y coordinate3.54

Derivative (Slope)

-1.000

\frac{dy}{dx} = -\frac{x}{y}

The Golden Rule

When taking the derivative of an implicit equation, you differentiate both sides with respect to $x$ .

The catch: Because $y$ is really just a hidden function of $x$ , every time you take the derivative of a $y$ term, you must multiply it by $dy/dx$ (This is just the Chain Rule!).

Normal (x)

Differentiating $x$ terms is business as usual.

\frac{d}{dx}(x^2) = 2x

Implicit (y)

Differentiating $y$ requires the chain rule attachment.

\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}

Solving the Circle

Let's solve the math behind the interactive lab above:

1. Original Equation

x^2 + y^2 = 25

2. Differentiate Both Sides

2x + 2y \frac{dy}{dx} = 0

3. Isolate dy/dx

2y \frac{dy}{dx} = -2x

\frac{dy}{dx} = -\frac{x}{y}

Notice that the final answer has both $x$ and $y$ in it. That's totally normal for implicit differentiation! It just means to find the slope, you need to know the specific point $(x, y)$ on the graph.