Explicit: Differentiate directly. Implicit: Differentiate with respect to x, using chain rule for y terms, then solve for dy/dx. Explicit is easier when possible.

Horizontal: dy/dx = 0. Vertical: dy/dx is undefined. Horizontal indicates a local extremum or inflection point. Vertical indicates a cusp or sharp turn.

Solving for x: expresses x as a function of y. Solving for y: expresses y as a function of x. The choice depends on which is easier and what information you need.

Explicit: Chain rule applied when differentiating a composite function directly. Implicit: Chain rule always applied when differentiating y terms with respect to x.

The domain of the explicit solution can be more restricted than the implicit equation due to square roots or other restrictions.

Explicit: Rates are usually directly given. Implicit: Rates are related through an equation, requiring implicit differentiation with respect to time.

Explicit: Quotient rule used when differentiating a quotient of functions directly. Implicit: Quotient rule may be needed when solving for $\frac{dy}{dx}$ after implicit differentiation.

Explicit: Often simpler to visualize directly. Implicit: Can represent more complex relationships, but may require more analysis to graph.

Implicit: Differentiate $x^2+y^2=r^2$ directly. Explicit: Solve for y first, then differentiate. Implicit is often easier.

Explicit: Find dy/dx directly, then use point-slope form. Implicit: Find dy/dx implicitly, then use point-slope form. Implicit is necessary if you cannot solve for y.

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 $y = \pm r$.