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.

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

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

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

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

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

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

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

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

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

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

Explicit functions have 'y' isolated. Implicit functions have 'x' and 'y' mixed, requiring implicit differentiation.

It reveals explicit functions representing parts of the implicit relation and helps determine domain and range.

Positive slope: 'y' increases as 'x' increases. Negative slope: 'y' decreases as 'x' increases. Zero slope: horizontal tangent. Undefined slope: vertical tangent.

The function has a local maximum or minimum, or a point of inflection where the slope is momentarily zero.

The derivative is undefined, often indicating a cusp or a point where the function changes direction sharply.

Differentiate both sides with respect to x, apply the chain rule where necessary, and solve for dy/dx.

The domain of the resulting explicit function(s) may be narrower than the original implicit equation's allowed x-values.

Find dy/dx using implicit differentiation, evaluate at the given point to find the slope, and use the point-slope form of a line.

Related rates problems involve finding the rate at which one quantity is changing by relating it to other quantities whose rates of change are known, often using implicit differentiation.

When differentiating terms involving 'y' with respect to 'x', the chain rule requires multiplying by $\frac{dy}{dx}$ because 'y' is a function of 'x'.