Glossary

Chain Rule

Criticality: 3

A fundamental derivative rule used to find the derivative of a composite function, where one function is nested within another.

Example:

To find the derivative of $f(x) = (3x^2 + 1)^5$ , you must apply the Chain Rule, differentiating the outer power function and then multiplying by the derivative of the inner polynomial.

Implicit Differentiation

Criticality: 3

A technique used to find the derivative of a function that is not explicitly defined in terms of one variable, often involving equations where variables are mixed.

Example:

When working with equations like $x^2 + y^2 = 25$ (a circle), you use implicit differentiation to find $\frac{dy}{dx}$ or $\frac{dy}{dt}$ .

Known Quantities

Criticality: 2

Specific values or rates of change that are explicitly provided within the problem statement of a related rates scenario.

Example:

In a problem about a conical tank draining, the initial height of the water and the rate at which the water level is dropping are typically known quantities.

Modeling the relationship between quantities

Criticality: 3

The crucial step of establishing a mathematical formula or equation that accurately connects all the relevant variables in a related rates problem.

Example:

For a problem involving a melting ice cube, modeling the relationship between quantities would involve using the formula for the volume of a cube, $V = s^3$ .

Product Rule

Criticality: 2

A derivative rule used to find the derivative of a function that is the product of two or more differentiable functions.

Example:

If you need to differentiate $g(x) = x^2 \sin(x)$ , the Product Rule is essential to correctly find $g'(x)$ .

Pythagorean Theorem

Criticality: 2

A fundamental geometric theorem stating that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$).

Example:

In the classic ladder problem, where a ladder slides down a wall, the ladder, wall, and ground form a right triangle, making the Pythagorean Theorem the perfect formula to relate their changing lengths.

Quantity of Time (with respect to time)

Criticality: 3

In related rates problems, derivatives are typically calculated with respect to time, meaning how a quantity changes over a period of time.

Example:

If a car's position is given by $x(t)$ , its velocity is the rate of change of position with respect to time, denoted as $\frac{dx}{dt}$ .

Quotient Rule

Criticality: 1

A derivative rule used to find the derivative of a function that is expressed as the ratio of two differentiable functions.

Example:

To find the derivative of $h(x) = \frac{\cos(x)}{e^x}$ , you would apply the Quotient Rule.

Related Rates

Criticality: 3

A calculus application where derivatives are used to find the rate at which a quantity changes in relation to the rates of change of other related quantities.

Example:

When a spherical balloon is being inflated, related rates can help determine how fast its radius is increasing given the rate at which its volume is growing.

Unknown Quantities

Criticality: 2

The specific value or rate of change that the related rates problem asks you to determine.

Example:

If a problem asks 'how fast is the shadow lengthening?', then the rate of change of the shadow's length is the unknown quantity you need to solve for.