$x^2 + y^2 = r^2$

Differentiate both sides of the equation with respect to x, remembering the chain rule for terms involving y. Then, solve for $\frac{dy}{dx}$.

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

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

$y = \pm \frac{1}{2}\sqrt{16 - x^2}$

$m = \frac{dy}{dx}$

$\frac{d}{dx}f(x,y) = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}$

$\frac{d}{dx}(y^n) = ny^{n-1} \frac{dy}{dx}$

$y - y_0 = \frac{dy}{dx}|_{(x_0, y_0)} (x - x_0)$

$y = \pm \sqrt{1-x^2}$

An equation where x and y are related, but y is not explicitly isolated.

A function where y is isolated on one side of the equation, expressed directly in terms of x.

The rate of change of y with respect to x (dy/dx) at that point, representing the tangent line's steepness.

An interval where the rate of change of x with respect to y is zero (dx/dy = 0), resulting in a slope of zero.

An interval where the rate of change of y with respect to x is undefined, resulting in an undefined slope.

Isolating y in terms of x, which may result in one or more explicit functions representing parts of the original implicit relation.

The set of all possible x-values for which the function is defined.

The set of all possible y-values that the function can take.

A method used to find the derivative of an implicitly defined function by differentiating both sides of the equation with respect to x, treating y as a function of x.

The graph is the set of all ordered pairs (x, y) that satisfy the equation.

As x increases, y also increases. The function is increasing at that point.

As x increases, y decreases. The function is decreasing at that point.

The derivative is zero, indicating a local maximum, local minimum, or a point of inflection.

The derivative is undefined, often indicating a cusp or a sharp turn in the graph.

Domain: the set of all x-values covered by the graph. Range: the set of all y-values covered by the graph.

If replacing x with -x or y with -y leaves the equation unchanged, the graph has symmetry about the y-axis or x-axis, respectively.

Concave up: second derivative is positive. Concave down: second derivative is negative.

Increasing: the graph has a positive slope. Decreasing: the graph has a negative slope.

The function repeats its values in regular intervals. This can be confirmed by checking if $f(x+T) = f(x)$ for some constant T.

The graph is a circle. The derivative gives the slope of the tangent line at any point on the circle. The slope will be undefined at $x = \pm r$, and zero at $y = \pm r$.