A function defined by an equation with multiple variables on the same side, not explicitly solved for one variable.

A technique to find the derivative of an implicit function by differentiating both sides of the equation with respect to a variable.

Points where the derivative $\frac{dy}{dx}$ is either 0 or undefined, indicating potential minima or maxima.

A point where the concavity of a function changes, and the second derivative $f''(x) = 0$ or is undefined.

Potential critical points where the tangent line is horizontal, possibly indicating a local minimum or maximum.

Potential critical points where the tangent line is vertical, possibly indicating a cusp or vertical tangent.

A method to determine if a critical point is a local minimum or maximum by analyzing the sign change of the first derivative around that point.

A method to determine the concavity of a function and identify points of inflection using the sign of the second derivative.

The function is concave up.

The function is concave down.

Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$.

1. Find $\frac{dy}{dx}$. 2. Set $\frac{dy}{dx} = 0$ and undefined. 3. Solve for x and y.

1. Find $\frac{dy}{dx}$. 2. Evaluate $\frac{dy}{dx}$ around $(0, 4)$. 3. Check for sign change.

1. Find $\frac{dy}{dx}$. 2. Use chain rule: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$. 3. Substitute values.

1. Find $\frac{dy}{dx}$. 2. Find $\frac{d^2y}{dx^2}$. 3. Determine intervals where $\frac{d^2y}{dx^2}$ is positive or negative.

1. Find $f''(x)$. 2. Set $f''(x) = 0$ and solve for $x$. 3. Check for concavity change around these points.

1. Identify variables and rates. 2. Find equation relating variables. 3. Differentiate with respect to time. 4. Substitute and solve.

Use the first derivative test (sign change of $f'(x)$) or the second derivative test (sign of $f''(x)$).

Use the Pythagorean theorem: $a^2 + b^2 = c^2$.

1. Area formula: $A = \pi r^2$. 2. Differentiate with respect to time: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$.

Find $\frac{dy}{dx}$ using implicit differentiation, set it equal to 0 and undefined, and solve for x and y.

Positive $\frac{dy}{dx}$ means the function is increasing; negative means decreasing.

Positive $\frac{d^2y}{dx^2}$ means the function is concave up; negative means concave down.

Analyze the sign change of $\frac{dy}{dx}$ around a critical point. Positive to negative indicates a local maximum, negative to positive indicates a local minimum.

If $f''(x) > 0$, the function is concave up. If $f''(x) < 0$, the function is concave down.

It marks a change in the concavity of the function, where the second derivative changes sign.

Find $\frac{dy}{dx}$ and determine the intervals where it is positive (increasing) or negative (decreasing).

Find $\frac{d^2y}{dx^2}$ and determine the intervals where it is positive (concave up) or negative (concave down).

It relates the rates of change of different variables with respect to time, allowing us to find $\frac{dy}{dt}$ given $\frac{dx}{dt}$ and $\frac{dy}{dx}$.

It helps visualize the relationships between variables and identify the equation that relates them.