Glossary

Chain Rule

Criticality: 3

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

Example:

To differentiate $f(x) = \sin(x^2)$ , use the Chain Rule to get $f'(x) = \cos(x^2) \cdot 2x$ .

Composite Functions

Criticality: 3

A function formed by applying one function to the results of another function, often written as $f(g(x))$, where $g(x)$ is the inner function and $f(x)$ is the outer function.

Example:

$f(x) = \sin(x^2)$ is a composite function where the outer function is sine and the inner function is $x^2$ , requiring the Chain Rule for differentiation.

Concave Up/Concave Down

Criticality: 3

Describes the curvature of a function's graph; concave up means the graph opens upwards (like a cup holding water), while concave down means it opens downwards (like a cup spilling water).

Example:

A function is concave up when its second derivative is positive, such as $f(x)=x^2$ for all $x$ .

Concavity

Criticality: 3

Describes the direction in which the graph of a function opens; it is concave up if it opens upwards and concave down if it opens downwards, determined by the sign of the second derivative.

Example:

A parabola $f(x) = x^2$ is concave up everywhere because its second derivative $f''(x) = 2$ is always positive.

Critical Points

Criticality: 3

Points in the domain of a function where the first derivative is either zero or undefined, which are candidates for relative extrema (minima or maxima).

Example:

For $f(x) = x^3 - 3x$ , the critical points are found by setting $f'(x) = 3x^2 - 3 = 0$ , which gives $x = \pm 1$ .

First Derivative

Criticality: 3

The derivative of a function, denoted as f'(x), which represents the instantaneous rate of change or slope of the function at any given point.

Example:

For $f(x) = x^2$ , the first derivative $f'(x) = 2x$ tells us the slope of the parabola at any x-value.

Higher-Order Derivatives

Criticality: 3

Derivatives of derivatives, found by repeatedly differentiating a function. They provide deeper insights into a function's behavior beyond its immediate rate of change.

Example:

If a function describes position, its first derivative is velocity, and its second derivative is acceleration; both are examples of higher-order derivatives.

Increasing/Decreasing Intervals

Criticality: 3

Intervals where the function's value is consistently rising (increasing) or falling (decreasing), determined by the sign of the first derivative.

Example:

If $f'(x) > 0$ on $(a, b)$ , then $f(x)$ is increasing on that interval, indicating the function's graph is going uphill.

Inflection Points

Criticality: 3

Points on the graph of a function where the concavity changes (from concave up to concave down or vice versa), typically where the second derivative is zero or undefined and changes sign.

Example:

For $f(x) = x^3$ , the inflection point is at $x=0$ because $f''(x)=6x$ changes sign from negative to positive there.

Natural Log Derivatives

Criticality: 2

The derivative of the natural logarithm function, $(\ln x)' = \frac{1}{x}$.

Example:

If $f(x) = \ln(3x)$ , its derivative using the Chain Rule and Natural Log Derivative is $f'(x) = \frac{1}{3x} \cdot 3 = \frac{1}{x}$ .

Power Rule

Criticality: 3

A fundamental differentiation rule stating that if $f(x) = ax^n$, then its derivative is $f'(x) = nax^{n-1}$.

Example:

To find the derivative of $f(x) = 5x^3$ , apply the Power Rule to get $f'(x) = 3 \cdot 5x^{3-1} = 15x^2$ .

Product Rule

Criticality: 3

A rule for differentiating the product of two functions, $L(x) \cdot R(x)$, where the derivative is $L'(x)R(x) + L(x)R'(x)$.

Example:

To find the derivative of $f(x) = x^2 \cos(x)$ , apply the Product Rule: $f'(x) = (2x)\cos(x) + x^2(-\sin(x))$ .

Quotient Rule

Criticality: 3

A rule for differentiating the quotient of two functions, $\frac{N(x)}{D(x)}$, where the derivative is $\frac{D(x)N'(x) - N(x)D'(x)}{(D(x))^2}$.

Example:

To differentiate $f(x) = \frac{\sin(x)}{x}$ , use the Quotient Rule: $f'(x) = \frac{x\cos(x) - \sin(x)(1)}{x^2}$ .

Relative Minima or Maxima

Criticality: 3

Points on a function where the function changes from decreasing to increasing (relative minimum) or increasing to decreasing (relative maximum), often occurring where the first derivative is zero or undefined.

Example:

For $f(x) = x^2$ , the relative minimum occurs at $x=0$ where $f'(x)=2x=0$ .

Second Derivative

Criticality: 3

The derivative of the first derivative, denoted as f''(x), which provides information about the concavity of the original function and helps identify inflection points.

Example:

If $f(x)$ represents position, the second derivative $f''(x)$ represents acceleration, indicating how the velocity is changing.

Slope

Criticality: 3

The measure of the steepness and direction of a line or curve at a specific point, given by the value of the first derivative.

Example:

The slope of the tangent line to $f(x) = x^3$ at $x=1$ is found by evaluating $f'(1) = 3(1)^2 = 3$ .

Trigonometric Derivatives

Criticality: 3

The derivatives of trigonometric functions, such as $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, and $( an x)' = \sec^2 x$.

Example:

The trigonometric derivative of $an(x)$ is $\sec^2(x)$ , which is often used in combination with the Chain Rule.

Glossary

Chain Rule

Criticality: 3

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

Example:

To differentiate $f(x) = \sin(x^2)$ , use the Chain Rule to get $f'(x) = \cos(x^2) \cdot 2x$ .

Composite Functions

Criticality: 3

A function formed by applying one function to the results of another function, often written as $f(g(x))$, where $g(x)$ is the inner function and $f(x)$ is the outer function.

Example:

$f(x) = \sin(x^2)$ is a composite function where the outer function is sine and the inner function is $x^2$ , requiring the Chain Rule for differentiation.

Concave Up/Concave Down

Criticality: 3

Describes the curvature of a function's graph; concave up means the graph opens upwards (like a cup holding water), while concave down means it opens downwards (like a cup spilling water).

Example:

A function is concave up when its second derivative is positive, such as $f(x)=x^2$ for all $x$ .

Concavity

Criticality: 3

Describes the direction in which the graph of a function opens; it is concave up if it opens upwards and concave down if it opens downwards, determined by the sign of the second derivative.

Example:

A parabola $f(x) = x^2$ is concave up everywhere because its second derivative $f''(x) = 2$ is always positive.

Critical Points

Criticality: 3

Points in the domain of a function where the first derivative is either zero or undefined, which are candidates for relative extrema (minima or maxima).

Example:

For $f(x) = x^3 - 3x$ , the critical points are found by setting $f'(x) = 3x^2 - 3 = 0$ , which gives $x = \pm 1$ .

First Derivative

Criticality: 3

The derivative of a function, denoted as f'(x), which represents the instantaneous rate of change or slope of the function at any given point.

Example:

For $f(x) = x^2$ , the first derivative $f'(x) = 2x$ tells us the slope of the parabola at any x-value.

Higher-Order Derivatives

Criticality: 3

Derivatives of derivatives, found by repeatedly differentiating a function. They provide deeper insights into a function's behavior beyond its immediate rate of change.

Example:

If a function describes position, its first derivative is velocity, and its second derivative is acceleration; both are examples of higher-order derivatives.

Increasing/Decreasing Intervals

Criticality: 3

Intervals where the function's value is consistently rising (increasing) or falling (decreasing), determined by the sign of the first derivative.

Example:

If $f'(x) > 0$ on $(a, b)$ , then $f(x)$ is increasing on that interval, indicating the function's graph is going uphill.

Inflection Points

Criticality: 3

Points on the graph of a function where the concavity changes (from concave up to concave down or vice versa), typically where the second derivative is zero or undefined and changes sign.

Example:

For $f(x) = x^3$ , the inflection point is at $x=0$ because $f''(x)=6x$ changes sign from negative to positive there.

Natural Log Derivatives

Criticality: 2

The derivative of the natural logarithm function, $(\ln x)' = \frac{1}{x}$.

Example:

If $f(x) = \ln(3x)$ , its derivative using the Chain Rule and Natural Log Derivative is $f'(x) = \frac{1}{3x} \cdot 3 = \frac{1}{x}$ .

Power Rule

Criticality: 3

A fundamental differentiation rule stating that if $f(x) = ax^n$, then its derivative is $f'(x) = nax^{n-1}$.

Example:

To find the derivative of $f(x) = 5x^3$ , apply the Power Rule to get $f'(x) = 3 \cdot 5x^{3-1} = 15x^2$ .

Product Rule

Criticality: 3

A rule for differentiating the product of two functions, $L(x) \cdot R(x)$, where the derivative is $L'(x)R(x) + L(x)R'(x)$.

Example:

To find the derivative of $f(x) = x^2 \cos(x)$ , apply the Product Rule: $f'(x) = (2x)\cos(x) + x^2(-\sin(x))$ .

Quotient Rule

Criticality: 3

A rule for differentiating the quotient of two functions, $\frac{N(x)}{D(x)}$, where the derivative is $\frac{D(x)N'(x) - N(x)D'(x)}{(D(x))^2}$.

Example:

To differentiate $f(x) = \frac{\sin(x)}{x}$ , use the Quotient Rule: $f'(x) = \frac{x\cos(x) - \sin(x)(1)}{x^2}$ .

Relative Minima or Maxima

Criticality: 3

Example:

For $f(x) = x^2$ , the relative minimum occurs at $x=0$ where $f'(x)=2x=0$ .

Second Derivative

Criticality: 3

The derivative of the first derivative, denoted as f''(x), which provides information about the concavity of the original function and helps identify inflection points.

Example:

If $f(x)$ represents position, the second derivative $f''(x)$ represents acceleration, indicating how the velocity is changing.

Slope

Criticality: 3

The measure of the steepness and direction of a line or curve at a specific point, given by the value of the first derivative.

Example:

The slope of the tangent line to $f(x) = x^3$ at $x=1$ is found by evaluating $f'(1) = 3(1)^2 = 3$ .

Trigonometric Derivatives

Criticality: 3

The derivatives of trigonometric functions, such as $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, and $( an x)' = \sec^2 x$.

Example:

The trigonometric derivative of $an(x)$ is $\sec^2(x)$ , which is often used in combination with the Chain Rule.