Glossary

Algebraic Simplification

Criticality: 2

The process of rewriting an algebraic expression in a simpler, more compact, or more manageable form, often by combining like terms, factoring, or canceling common factors.

Example:

After finding the derivative of $\frac{x^2+x-2}{x^3+6}$ , you'll need to use algebraic simplification to combine terms like $2x^4 - 3x^4$ into $-x^4$ .

Denominator

Criticality: 1

In a fraction, the **denominator** is the bottom part, representing the divisor or the quantity by which the numerator is divided.

Example:

In the Quotient Rule formula, the denominator $g(x)$ is squared in the final expression.

Derivative

Criticality: 3

The **derivative** of a function measures the instantaneous rate of change of the function with respect to its independent variable, often interpreted as the slope of the tangent line at any given point.

Example:

If a car's position is given by $s(t)$ , its derivative $s'(t)$ gives the car's instantaneous velocity.

Differentiable

Criticality: 2

A function is **differentiable** at a point if its derivative exists at that point, meaning it has a well-defined tangent line and no sharp corners, cusps, or vertical tangents.

Example:

While $f(x) = x^2$ is differentiable everywhere, $g(x) = |x-3|$ is not differentiable at $x=3$ due to a sharp corner.

Numerator

Criticality: 1

In a fraction, the **numerator** is the top part, representing the dividend or the quantity being divided.

Example:

When applying the Quotient Rule to $\frac{e^x}{1+x^2}$ , $e^x$ is the numerator.

Quotient Rule

Criticality: 3

A formula used to find the derivative of a function that is expressed as the ratio of two differentiable functions, $\frac{f(x)}{g(x)}$.

Example:

To find the rate at which the concentration of a drug in the bloodstream changes over time, given by $C(t) = \frac{t}{t^2+1}$ , you would apply the Quotient Rule.

Rational Functions

Criticality: 2

Functions that can be expressed as the ratio of two polynomial functions, where the denominator is not zero.

Example:

The function representing the average cost per item, $A(x) = \frac{C(x)}{x}$ , where $C(x)$ is the total cost, is often a rational function.

Trigonometric Identity

Criticality: 2

An equation involving trigonometric functions that is true for all values of the variables for which the functions are defined, such as $\sin^2(x) + \cos^2(x) = 1$.

Example:

When differentiating $\frac{\sin(x)}{1+\cos(x)}$ , the expression $\cos^2(x) + \sin^2(x)$ can be simplified to 1 using a fundamental Trigonometric Identity.