Glossary

Chain Rule

Criticality: 3

A rule for differentiating composite functions, stating that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Example:

To find the derivative of $y = \sin(x^3)$ , you must apply the Chain Rule, resulting in $3x^2 \cos(x^3)$ .

Curve Sketching

Criticality: 2

The process of using calculus concepts like derivatives (first and second) to analyze the behavior of a function and accurately draw its graph.

Example:

Using the first derivative to find increasing/decreasing intervals and the second derivative for concavity are essential steps in Curve Sketching.

Derivative

Criticality: 3

The instantaneous rate of change of a function with respect to its independent variable, representing the slope of the tangent line to the function's graph at a given point.

Example:

Finding the derivative of a position function gives you the velocity, telling you how fast something is moving at any exact moment.

Implicit Differentiation

Criticality: 3

A technique used to differentiate functions where the dependent variable cannot be easily expressed explicitly in terms of the independent variable, by differentiating both sides of an equation with respect to the independent variable.

Example:

When dealing with equations like $x^2 + y^2 = 25$ (a circle), you use Implicit Differentiation to find $\frac{dy}{dx}$ without solving for $y$ first.

Inverse Function Derivative Formula

Criticality: 2

A general formula used to find the derivative of an inverse function, stating that $\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$.

Example:

While you'll memorize specific inverse trig derivatives, the Inverse Function Derivative Formula is how mathematicians originally derived them, like showing why $\frac{d}{dx}[\ln(x)] = \frac{1}{x}$ from $e^x$ .

Inverse Trigonometric Functions

Criticality: 3

Functions that 'undo' the regular trigonometric functions, returning an angle whose sine, cosine, etc., is a given value.

Example:

If you know $\sin( heta) = 0.5$ , then $heta = \sin^{-1}(0.5) = \pi/6$ . Understanding Inverse Trigonometric Functions is key to solving for angles in calculus problems.

L'Hôpital's Rule

Criticality: 2

A rule used to evaluate limits of indeterminate forms (like 0/0 or \infty/\infty) by taking the derivatives of the numerator and denominator separately.

Example:

If you encounter $\lim_{x \to 0} \frac{\sin(x)}{x}$ , you can apply L'Hôpital's Rule to quickly find the limit is 1.

Optimization

Criticality: 2

The process of finding the maximum or minimum value of a function, often used to solve real-world problems involving maximizing profit or minimizing cost.

Example:

To find the dimensions of a cylindrical can that minimize the amount of material used for a given volume, you would solve an Optimization problem.

Product Rule

Criticality: 3

A rule for differentiating the product of two functions, stating that if $h(x) = f(x)g(x)$, then $h'(x) = f'(x)g(x) + f(x)g'(x)$.

Example:

When differentiating $x \cdot \sin^{-1}(x)$ , you'll need to use the Product Rule to correctly find $h'(x)$ .

Related Rates

Criticality: 2

Problems that involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known.

Example:

A classic Related Rates problem involves determining how fast the water level is rising in a conical tank as water is poured in.

Tangent Line

Criticality: 3

A straight line that touches a curve at a single point and has the same slope as the curve at that point.

Example:

Finding the equation of the Tangent Line to a function at a specific point is a common application of derivatives, often appearing in FRQs.