Glossary

Chain Rule

Criticality: 2

A calculus rule used to find the derivative of a composite function, which is a function within another function.

Example:

When calculating the rate of change of the volume of a balloon as it inflates, where volume depends on radius and radius depends on time, you would apply the Chain Rule to link these rates.

Chain Rule

Criticality: 3

A crucial differentiation rule used to find the derivative of a composite function. It states that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Example:

To differentiate $y = (3x^2 + 1)^5$ , you must apply the Chain Rule, treating $3x^2 + 1$ as the inner function.

Chain Rule

Criticality: 3

A fundamental rule in calculus used to differentiate composite functions. It states that the derivative of $f(g(x))$ is $f'(g(x)) \cdot g'(x)$.

Example:

When differentiating $y = (2x+1)^3$ , you apply the chain rule by differentiating the outer function (power of 3) and multiplying by the derivative of the inner function ( $2x+1$ ).

Chain Rule

Criticality: 3

A rule used to find the derivative of a composite function, stating that the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

Example:

To differentiate $y = \cos(4x^2)$ , you would use the Chain Rule to handle the function within a function.

Chain Rule

Criticality: 3

A differentiation rule used to find the derivative of composite functions. If $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$.

Example:

To differentiate $f(x) = \sin(x^2)$ , you apply the chain rule, treating $\sin(u)$ as the outer function and $x^2$ as the inner function.

Composite Function

Criticality: 2

A function formed by applying one function to the result of another function, often written as $f(g(x))$.

Example:

The function $y = e^{\sin(x)}$ is a composite function because the sine function is 'inside' the exponential function.

Constant Function (Differentiation)

Criticality: 2

A function of the form $f(x) = c$, where c is a constant, and its derivative is always zero.

Example:

If $g(x) = 100$ , its derivative $g'(x) = 0$ , as a Constant Function has no rate of change.

Constant Multiple Rule

Criticality: 3

States that the derivative of a constant times a function is the constant multiplied by the derivative of the function.

Example:

If $h(x) = 7x^3$ , the Constant Multiple Rule gives $h'(x) = 7 imes (3x^2) = 21x^2$ .

Constant Multiple Rule

Criticality: 2

A differentiation rule stating that the derivative of a constant times a function is the constant multiplied by the derivative of the function. If $c$ is a constant, then $\frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)]$.

Example:

To differentiate $y = 5x^2$ , you use the constant multiple rule to get $5 \cdot (2x) = 10x$ .

Cosecant (csc)

Criticality: 2

A reciprocal trigonometric function defined as the reciprocal of the sine function, $\csc x = \frac{1}{\sin x}$.

Example:

In optics, the path of light might involve angles where the cosecant function helps describe the inverse of the sine of the angle of incidence.

Cosine function

Criticality: 3

A fundamental trigonometric function, denoted as $\cos x$, which relates an angle of a right-angled triangle to the ratio of the length of the adjacent side to the length of the hypotenuse.

Example:

The horizontal displacement of a mass on a spring oscillating back and forth can be described by a Cosine function.

Cotangent (cot)

Criticality: 2

A reciprocal trigonometric function defined as the reciprocal of the tangent function, $\cot x = \frac{1}{\tan x}$ or $\frac{\cos x}{\sin x}$.

Example:

In surveying, the cotangent of an angle might be used to determine horizontal distances based on vertical measurements, as it relates the adjacent side to the opposite side.

Definition of a Derivative

Criticality: 2

The formal definition of a derivative using limits, expressed as $f'(x) = \lim_{h o 0} \frac{f(x+h) - f(x)}{h}$. This is also known as differentiation from first principles.

Example:

Using the Definition of a Derivative, one can rigorously prove that the derivative of $f(x) = x^2$ is $f'(x) = 2x$ .

Derivative

Criticality: 3

The result of differentiation, representing the instantaneous rate of change of a function with respect to its independent variable.

Example:

If a car's position is described by $s(t) = t^2$ , its velocity, which is the derivative $s'(t) = 2t$ , tells us how fast it's moving at any given time.

Derivative

Criticality: 3

A measure of how a function's output changes with respect to a change in its input. It represents the instantaneous rate of change or the slope of the tangent line to the function's graph.

Example:

If a car's position is described by $s(t) = t^3$ , its velocity at any time $t$ is given by the Derivative $s'(t) = 3t^2$ .

Derivative

Criticality: 3

The derivative of a function measures the instantaneous rate of change of the function with respect to its independent variable. It represents the slope of the tangent line to the function's graph 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.

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 any given point.

Example:

The derivative of a position function gives you the instantaneous velocity of an object.

Derivative of cot x

Criticality: 3

The rate of change of the cotangent function, given by $f'(x) = -\csc^2 x$. It can be derived using the quotient rule on $\cos x / \sin x$.

Example:

When analyzing the slope of a curve defined by $h(x) = \cot x$ , the instantaneous slope at any point $x$ is $h'(x) = -\csc^2 x$ .

Derivative of csc x

Criticality: 3

The rate of change of the cosecant function, given by $f'(x) = -\cot x \csc x$. It can be derived using the quotient rule on $1/\sin x$.

Example:

If a physical quantity's behavior is modeled by $f(x) = \csc x$ , its instantaneous rate of change would be $f'(x) = -\cot x \csc x$ .

Derivative of sec x

Criticality: 3

The rate of change of the secant function, given by $f'(x) = \tan x \sec x$. It can be derived using the quotient rule on $1/\cos x$.

Example:

In engineering, if a structural deflection is described by $g(x) = \sec x$ , the rate at which this deflection changes with respect to $x$ is given by $g'(x) = \tan x \sec x$ .

Derivative of tan kx

Criticality: 3

The derivative of the tangent function when its argument is scaled by a constant $k$. It is found using the chain rule and is given by $f'(x) = k \sec^2 kx$.

Example:

If a wave's amplitude is described by $A(t) = \tan(2t)$ , its rate of change of amplitude is $A'(t) = 2 \sec^2(2t)$ , illustrating the derivative of tan kx.

Derivative of tan x

Criticality: 3

The rate of change of the tangent function with respect to its variable. It is given by the formula $f'(x) = \sec^2 x$.

Example:

If you're modeling the slope of a ramp that changes according to $f(x) = \tan x$ , the instantaneous steepness at any point $x$ is given by $f'(x) = \sec^2 x$ .

Difference Rule

Criticality: 3

States that the derivative of a difference of functions is the difference of their individual derivatives.

Example:

For $g(x) = x^4 - 5x^2$ , the Difference Rule means $g'(x) = 4x^3 - 10x$ .

Differentiating $\ln kx$

Criticality: 2

The process of finding the derivative of a natural logarithmic function where the argument is a constant multiple of $x$, $\ln kx$, which simplifies to $\frac{1}{x}$ due to logarithm properties ($\ln kx = \ln k + \ln x$).

Example:

When analyzing a function like $f(x) = \ln(3x)$ , you can simplify it to $\ln 3 + \ln x$ before differentiating $\ln kx$ , leading to a derivative of $\frac{1}{x}$ .

Differentiating $\ln x$

Criticality: 3

The process of finding the derivative of the natural logarithmic function $\ln x$, which results in $\frac{1}{x}$.

Example:

If the intensity of a sound is measured on a logarithmic scale, $L(I) = \ln I$ , then the rate of change of intensity with respect to sound level involves differentiating $\ln x$ .

Differentiating $a \ln x$

Criticality: 2

The process of finding the derivative of a natural logarithmic function multiplied by a constant, $a \ln x$, which results in $\frac{a}{x}$ by applying the constant multiple rule.

Example:

To find the rate of change of a quantity modeled by $y = 5 \ln x$ , you would use the rule for differentiating $a \ln x$ .

Differentiating $a e^{kx}$

Criticality: 3

The process of finding the derivative of an exponential function multiplied by a constant, $a e^{kx}$, which results in $a k e^{kx}$ by combining the constant multiple rule and the chain rule.

Example:

If the temperature of a cooling object is $T(t) = 20 + 80e^{-0.1t}$ , finding the rate of cooling involves differentiating $a e^{kx}$ for the exponential term.

Differentiating $a^x$

Criticality: 2

The process of finding the derivative of a general exponential function $a^x$, where $a$ is a positive constant, which results in $a^x \ln a$.

Example:

If an investment doubles every 10 years, its value might be $V(t) = V_0 \cdot 2^{t/10}$ . To find the instantaneous growth rate, you would apply the rule for differentiating $a^x$ .

Differentiating $a^{kx}$

Criticality: 2

The process of finding the derivative of a general exponential function with a linear exponent, $a^{kx}$, which results in $a^{kx} k \ln a$ by applying the chain rule.

Example:

To model the growth of a specific type of bacteria that triples every hour, $N(t) = N_0 \cdot 3^{2t}$ , finding the growth rate requires differentiating $a^{kx}$ .

Differentiating $e^x$

Criticality: 3

The process of finding the derivative of the natural exponential function $e^x$, which is uniquely equal to itself.

Example:

If a bacterial colony grows according to $N(t) = 100e^t$ , then the rate of growth is found by differentiating $e^t$ , resulting in $N'(t) = 100e^t$ .

Differentiating $e^{kx}$

Criticality: 3

The process of finding the derivative of an exponential function with a linear exponent, $e^{kx}$, which results in $k e^{kx}$ by applying the chain rule.

Example:

To find the rate of decay of a radioactive substance modeled by $A(t) = A_0 e^{-0.05t}$ , you would use the rule for differentiating $e^{kx}$ to get $A'(t) = -0.05 A_0 e^{-0.05t}$ .

Differentiating cos x

Criticality: 3

The process of finding the derivative of the cosine function, $\cos x$, which results in $-\sin x$.

Example:

To determine the rate of change of a wave's horizontal displacement described by $x(t) = \cos t$ , you perform Differentiating cos x to get $x'(t) = -\sin t$ .

Differentiating cos(kx)

Criticality: 3

The process of finding the derivative of a cosine function with a linear argument $kx$, which results in $-k \sin(kx)$ by applying the chain rule.

Example:

If you have $g(x) = \cos(7x)$ , Differentiating cos(kx) yields $g'(x) = -7 \sin(7x)$ .

Differentiating sin x

Criticality: 3

The process of finding the derivative of the sine function, $\sin x$, which results in $\cos x$.

Example:

If a particle's vertical position is given by $y(t) = \sin t$ , its vertical velocity is found by Differentiating sin x, yielding $y'(t) = \cos t$ .

Differentiating sin(kx)

Criticality: 3

The process of finding the derivative of a sine function with a linear argument $kx$, which results in $k \cos(kx)$ by applying the chain rule.

Example:

For a function $f(x) = \sin(4x)$ , Differentiating sin(kx) gives $f'(x) = 4 \cos(4x)$ .

Differentiation

Criticality: 3

The mathematical process of finding the derivative of a function, which represents the instantaneous rate of change of the function.

Example:

Through differentiation, we can determine the velocity of an object given its position function over time.

Differentiation

Criticality: 3

The mathematical process of finding the derivative of a function, which determines how a function's output changes in response to changes in its input.

Example:

The act of calculating $f'(x)$ from $f(x)$ is known as differentiation, a core operation in calculus.

Expansion (Algebraic)

Criticality: 2

The process of multiplying out terms in an algebraic expression, often involving brackets, to simplify it into a sum or difference of terms.

Example:

Before differentiating $f(x) = (x+1)(x-2)$ , you would use Expansion to rewrite it as $f(x) = x^2 - x - 2$ .

Exponent

Criticality: 1

A number or symbol placed above and to the right of another number or symbol, indicating the power to which the base is to be raised.

Example:

In the term $x^n$ , 'n' is the Exponent, determining how many times 'x' is multiplied by itself.

Exponential Function

Criticality: 3

A function where the independent variable appears as an exponent, typically of the form $f(x) = b^x$ or $f(x) = e^x$, where $e$ is Euler's number. It models growth or decay that is proportional to the current amount.

Example:

Population growth often follows an exponential function, like $P(t) = P_0 e^{kt}$ , where $P_0$ is the initial population.

Fractional Power

Criticality: 2

An exponent that is a fraction, indicating both a root and a power, such as $x^{1/2}$ for $\sqrt{x}$.

Example:

When differentiating $f(x) = \sqrt[3]{x^2}$ , you first rewrite it as $f(x) = x^{2/3}$ to apply the power rule to this Fractional Power.

Laws of Exponents

Criticality: 2

Rules governing the manipulation of expressions involving exponents, such as $x^a \cdot x^b = x^{a+b}$ or $\frac{x^a}{x^b} = x^{a-b}$.

Example:

To differentiate $g(x) = \frac{x^5}{x^2}$ , you first apply the Laws of Exponents to simplify it to $g(x) = x^3$ .

Laws of Logarithms

Criticality: 2

A set of rules that govern how logarithms can be manipulated, including product rule ($\ln(AB) = \ln A + \ln B$), quotient rule ($\ln(A/B) = \ln A - \ln B$), and power rule ($\ln(A^n) = n \ln A$).

Example:

To simplify $\ln(x^2/y)$ , you would apply the laws of logarithms to rewrite it as $2\ln x - \ln y$ .

Limit

Criticality: 2

In calculus, a limit is the value that a function or sequence approaches as the input or index approaches some specific value. It is foundational to the definition of a derivative.

Example:

The concept of a Limit is used to define instantaneous velocity as the average velocity over an infinitesimally small time interval.

Limit Definition of a Derivative

Criticality: 2

The formal definition of a derivative using limits, expressed as $f'(x) = \lim_{h o 0} \frac{f(x+h) - f(x)}{h}$.

Example:

While the power rule is faster, understanding the Limit Definition of a Derivative helps explain why the power rule works and is crucial for deriving other differentiation rules.

Linear Function (Differentiation)

Criticality: 2

A function of the form $f(x) = ax + b$, whose derivative is simply the constant coefficient 'a'.

Example:

The derivative of $f(x) = 5x - 3$ is $f'(x) = 5$ , illustrating the rule for differentiating a Linear Function.

Natural Logarithm

Criticality: 3

The logarithm to the base $e$ (Euler's number), denoted as $\ln x$. It is the inverse function of the exponential function $e^x$.

Example:

To solve $e^x = 10$ , you would use the natural logarithm: $x = \ln(10)$ .

Negative Power

Criticality: 2

An exponent that is a negative number, indicating the reciprocal of the base raised to the positive equivalent of that power.

Example:

To differentiate $f(x) = \frac{1}{x^2}$ , you rewrite it as $f(x) = x^{-2}$ to handle the Negative Power before applying the power rule.

Power Rule

Criticality: 3

A fundamental rule in calculus used to find the derivative of functions in the form of x raised to a power.

Example:

To differentiate $f(x) = x^5$ , you apply the Power Rule to get $f'(x) = 5x^4$ .

Product Rule

Criticality: 3

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

Example:

If you're modeling the power output of a solar panel as the product of its efficiency $E(t)$ and the incident light intensity $I(t)$ , you'd use the Product Rule to find the rate of change of power.

Product Rule

Criticality: 3

A fundamental rule in calculus used to find the derivative of a function that is the product of two other differentiable functions.

Example:

When finding the derivative of $f(x) = x^3 \cdot e^x$ , you must apply the Product Rule to correctly differentiate both parts.

Product Rule

Criticality: 3

A differentiation rule used to find the derivative of a function that is the product of two other differentiable functions. If $y = uv$, then $y' = u'v + uv'$.

Example:

To find the derivative of $f(x) = x^3 \cos x$ , you would use the product rule with $u = x^3$ and $v = \cos x$ .

Product of two functions

Criticality: 2

A mathematical expression where two distinct functions are multiplied together, such as $g(x) \cdot h(x)$.

Example:

The expression $\sin(x) \cdot \ln(x)$ represents a product of two functions, requiring the product rule for differentiation.

Quotient Rule

Criticality: 3

A fundamental calculus rule used to find the derivative of a function that is expressed as the division of two other differentiable functions.

Example:

To find the rate of change of a population density, where population is $P(t)$ and area is $A(t)$ , you would use the Quotient Rule on $D(t) = \frac{P(t)}{A(t)}$ .

Quotient Rule

Criticality: 3

A differentiation rule used to find the derivative of a function that is the ratio of two other differentiable functions. If $y = u/v$, then $y' = (u'v - uv')/v^2$.

Example:

To find the derivative of $f(x) = \frac{\sin x}{x^2}$ , you would apply the quotient rule with $u = \sin x$ and $v = x^2$ .

Reciprocal Trigonometric Functions

Criticality: 2

A set of trigonometric functions defined as the reciprocals of the primary trigonometric functions (sine, cosine, tangent). These include cosecant, secant, and cotangent.

Example:

When analyzing oscillations, you might encounter $\sec x$ or $\csc x$ as reciprocal trigonometric functions that describe inverse relationships to sine or cosine.

Secant (sec)

Criticality: 2

A reciprocal trigonometric function defined as the reciprocal of the cosine function, $\sec x = \frac{1}{\cos x}$.

Example:

When calculating the length of a hypotenuse in relation to an adjacent side, the secant function can be directly applied as the reciprocal of the cosine.

Sine function

Criticality: 3

A fundamental trigonometric function, denoted as $\sin x$, which relates an angle of a right-angled triangle to the ratio of the length of the opposite side to the length of the hypotenuse.

Example:

The height of a point on a spinning bicycle wheel above the ground can be modeled using a Sine function over time.

Sum Rule

Criticality: 3

States that the derivative of a sum of functions is the sum of their individual derivatives.

Example:

If $f(x) = x^3 + 2x$ , the Sum Rule allows you to differentiate each term separately to get $f'(x) = 3x^2 + 2$ .

Trigonometric Addition Formulae

Criticality: 2

Identities that allow the expansion of trigonometric functions of sums or differences of angles into expressions involving trigonometric functions of the individual angles.

Example:

When proving the derivative of $\cos x$ from first principles, the Trigonometric Addition Formulae are essential for expanding $\cos(x+h)$ .

Trigonometric Identities

Criticality: 2

Equations involving trigonometric functions that are true for every value of the variables for which both sides of the equation are defined. They are crucial for simplifying expressions.

Example:

Using the trigonometric identity $\sin^2 x + \cos^2 x = 1$ is essential for simplifying expressions like $\frac{\sin^2 x + \cos^2 x}{\cos^2 x}$ to $\sec^2 x$ .