Glossary

Chain Rule

Criticality: 3

A differentiation rule used for finding the derivative of composite functions, stating that the derivative is the derivative of the outer function (evaluated at the inner function) multiplied by the derivative of the inner function.

Example:

To find the derivative of $y = (3x+5)^4$ , you'd use the chain rule to get $4(3x+5)^3 \cdot 3 = 12(3x+5)^3$ .

Composite Functions

Criticality: 3

Functions within functions, where the output of one function serves as the input for another, creating a nested structure.

Example:

If $h(x) = \sqrt{\sin(x)}$ , then $h(x)$ is a composite function because $\sin(x)$ is inside the square root function.

Concavity

Criticality: 3

A property of a curve that describes its curvature; a function is concave up if its graph opens upwards and concave down if it opens downwards, determined by the sign of the second derivative.

Example:

A parabola opening upwards, like $y=x^2$ , exhibits concavity up, as its second derivative is positive.

Difference Quotient

Criticality: 1

An expression $\frac{f(b) - f(a)}{b - a}$ used to approximate the derivative of a function over a small interval, representing the average rate of change.

Example:

To estimate the instantaneous rate of change from a table of values, you might use the difference quotient between two closely spaced points.

Higher-Order Derivatives

Criticality: 2

Derivatives of derivatives, such as the second derivative ($f''(x)$), third derivative ($f'''(x)$), and so on, obtained by repeatedly differentiating the function.

Example:

If $f(x) = x^4$ , its first derivative is $4x^3$ , and its second derivative is $12x^2$ , which is an example of a higher-order derivative.

Implicit Differentiation

Criticality: 3

A technique used to find the derivative of a function where $y$ is not explicitly defined in terms of $x$, by differentiating all terms with respect to $x$ and applying the chain rule to terms involving $y$.

Example:

Finding $\frac{dy}{dx}$ for the equation $x^3 + y^3 = 6xy$ requires implicit differentiation because $y$ is not isolated.

Inner Function

Criticality: 3

In a composite function $f(g(x))$, the inner function is $g(x)$, which is evaluated first.

Example:

For $y = e^{x^3}$ , the inner function is $x^3$ , as it's the first operation applied to $x$ before the exponential.

Outer Function

Criticality: 3

In a composite function $f(g(x))$, the outer function is $f$, which operates on the result of the inner function.

Example:

Given $y = \ln(x^2 + 1)$ , the natural logarithm is the outer function that acts upon the polynomial $x^2 + 1$ .

Points of Inflection

Criticality: 2

Points on a curve where the concavity changes (from concave up to concave down or vice versa), typically occurring where the second derivative is zero or undefined.

Example:

For $f(x) = x^3$ , the origin $(0,0)$ is a point of inflection because the graph changes from concave down to concave up at that point.

Second Derivative

Criticality: 3

The derivative of the first derivative of a function, denoted as $f''(x)$ or $\frac{d^2y}{dx^2}$, which provides information about the concavity of the original function.

Example:

If $f'(x)$ tells you velocity, then the second derivative $f''(x)$ tells you acceleration, indicating how the velocity is changing.

Glossary

Chain Rule

Criticality: 3

Example:

To find the derivative of $y = (3x+5)^4$ , you'd use the chain rule to get $4(3x+5)^3 \cdot 3 = 12(3x+5)^3$ .

Composite Functions

Criticality: 3

Functions within functions, where the output of one function serves as the input for another, creating a nested structure.

Example:

If $h(x) = \sqrt{\sin(x)}$ , then $h(x)$ is a composite function because $\sin(x)$ is inside the square root function.

Concavity

Criticality: 3

A property of a curve that describes its curvature; a function is concave up if its graph opens upwards and concave down if it opens downwards, determined by the sign of the second derivative.

Example:

A parabola opening upwards, like $y=x^2$ , exhibits concavity up, as its second derivative is positive.

Difference Quotient

Criticality: 1

An expression $\frac{f(b) - f(a)}{b - a}$ used to approximate the derivative of a function over a small interval, representing the average rate of change.

Example:

To estimate the instantaneous rate of change from a table of values, you might use the difference quotient between two closely spaced points.

Higher-Order Derivatives

Criticality: 2

Derivatives of derivatives, such as the second derivative ($f''(x)$), third derivative ($f'''(x)$), and so on, obtained by repeatedly differentiating the function.

Example:

If $f(x) = x^4$ , its first derivative is $4x^3$ , and its second derivative is $12x^2$ , which is an example of a higher-order derivative.

Implicit Differentiation

Criticality: 3

Example:

Finding $\frac{dy}{dx}$ for the equation $x^3 + y^3 = 6xy$ requires implicit differentiation because $y$ is not isolated.

Inner Function

Criticality: 3

In a composite function $f(g(x))$, the inner function is $g(x)$, which is evaluated first.

Example:

For $y = e^{x^3}$ , the inner function is $x^3$ , as it's the first operation applied to $x$ before the exponential.

Outer Function

Criticality: 3

In a composite function $f(g(x))$, the outer function is $f$, which operates on the result of the inner function.

Example:

Given $y = \ln(x^2 + 1)$ , the natural logarithm is the outer function that acts upon the polynomial $x^2 + 1$ .

Points of Inflection

Criticality: 2

Points on a curve where the concavity changes (from concave up to concave down or vice versa), typically occurring where the second derivative is zero or undefined.

Example:

For $f(x) = x^3$ , the origin $(0,0)$ is a point of inflection because the graph changes from concave down to concave up at that point.

Second Derivative

Criticality: 3

The derivative of the first derivative of a function, denoted as $f''(x)$ or $\frac{d^2y}{dx^2}$, which provides information about the concavity of the original function.

Example:

If $f'(x)$ tells you velocity, then the second derivative $f''(x)$ tells you acceleration, indicating how the velocity is changing.