Average
Average, abbreviated as \(\large{ \bar x }\), is the mean value of a quantity over a specific interval or set of data. It represents the total sum ( such as \(\bar x = \dfrac{ x_1 + x_2 + x_3 }{ 3 } \;\) or ) of all values divided by the number of values or the duration of time considered. The average gives a single representative value that describes the overall trend of a varying quantity without showing its instant-by-instant changes. It is useful when precise variations are not required, and only a general idea of performance or behavior is needed.
Mean
Mean, abbreviated as \(\large{ \bar x }\), is a measure of central tendency that represents the average of a set of numbers or data points. It is found by adding together all the values in a dataset and then dividing the total by the number of values. Mathematically, it is expressed as the mean of a set of simple data \( \bar x = \dfrac{ \sum ( x_i + x_j ) }{ n } \), where \( x_i \) and \( x_j \) represents individual data point swithin a set. The mean provides a single representative value that summarizes the overall level or trend of the data. In physics and engineering, the mean value is often used to describe average quantities, such as mean velocity, mean current, or mean pressure, when a variable changes over time or space. It helps simplify complex data by giving a general value that reflects the typical or expected outcome of a system.
Change
Change, abbreviated as \(\large{ \Delta }\) (Greak symbol delta), is the difference between a final and an initial value of a quantity over a certain interval. It is expressed as. Change measures how much a variable increases or decreases over time, distance, or another parameter. For instance, in physics, \(\; \Delta v = v_2 - v_1\) describes the change in velocity over a given time period. Unlike the average, which gives a ratio or mean value, change shows the total amount of variation in a quantity between two points or conditions.
Differential
Differential, abbreviated as \(\large{ d }\), represents an infinitesimally small change in a quantity and is used to describe how one variable changes in relation to another. The differential is written using the symbol and placed before a variable, such as \(dx\) or \(dy\), and serves as the basis for derivatives and integrals. It helps define the derivative \( \dfrac{dx}{dy} \) as the ratio of two differentials and appears in integrals as the element of change being summed. In physics and engineering, differentials are used to express very small variations of physical quantities, such as \(d\rho\) for pressure or \(dV\) for volume, allowing equations to accurately model continuous processes. Overall, a differential provides a precise way to analyze how systems evolve through tiny, incremental changes.
Derivative
In mathematics, a derivative is a concept that measures how a function changes at a specific point, essentially representing the instantaneous rate of change of one quantity relative to another. For a function \(\;y = f(x)\), the derivative, denoted as \(\;f'(x)\), \(\;dy / dx\), or \(\; d / dx [f(x)] \), is defined as the limit of the difference quotient \(\; [ f(x+h) - f(x)] / h \) as \(\;h\) approaches zero, provided that limit exists. Geometrically, the derivative at a point gives the slope of the tangent line to the curve at that point, while in applied contexts it describes quantities like velocity (the derivative of position with respect to time), acceleration (the derivative of velocity), or marginal cost in economics.
In science, particularly physics, engineering, chemistry, and biology, the derivative appears whenever we need to express rates of change. For example, in physics, if an object’s position is given as a function of time \(\;s(t)\), then its velocity is \(\;v(t) = ds / dt\) (the first derivative) and its acceleration is \(\; a(t) = dv / dt = d^2 s / d t^2\) (the second derivative). In thermodynamics, the derivative of internal energy with respect to temperature gives heat capacity; in chemical kinetics, the derivative of concentration with respect to time gives the rate of reaction; and in biology, population growth models often use derivatives to express how populations change over time (e.g., the logistic equation). In all these fields, derivatives allow scientists to move from total or accumulated quantities (distance traveled, total heat added, total molecules reacted) to instantaneous rates, which are often the physically or experimentally meaningful quantities. Thus, the derivative is one of the most powerful and universal tools for describing dynamic processes in both pure mathematics and throughout the natural sciences.
- A derivative is a math expression of a rate of change. Any rate of change of a moving object related to the other object can be expressed as a derivative.
- A derivative is a rate of change, which is the slope of a graph in geometric terms. \({\large slope = \frac{ change\; in \;y }{ change\; in \; x } }\)
- The mark \('\) means derivative of, and \(f\) and \(g\) are functions.
- The slope of a constant value is always 0.
Differential Equations Terms
Differential equation, abbreviated as \(\Delta\) (Greek symbol Delta), is an equation which contains one or more terms and the derivatives of one variable with respect to the other variable. The rate at which this moving object changes relative to the other object can be expressed as the derivative of that initial mathematical equation.
Differential Equations - These are problems that require the determination of a function satisfying an equation containing one or more derivatives of the unknown function , such as unknown function \(y\) or \(y(x)\) and its dreivatives \(\frac{ dy }{ dx }\) or \(\dfrac{ d^2y }{ dx^2 }\). Since derivatives represent the rate of change, differential equations essentially relate to its rate of change, or the rate of change of mulyiple quantities.
Ordinary Differential Equations - The unknown function in the equation only depends on one independent variable; as a result only ordinary derivatives appear in the equation.
Partial Differential Equations - The unknown function depends on more than one independent variable; as a result partial derivatives appear in the equation, such as \(\frac{ \partial y }{ \partial x }\) or \(\dfrac{ \partial^2y }{ \partial x^2 }\), which shows the change with respect to one variable while the other is held constant.
Order of Differential Equations - The order of a differential equation (partial or ordinary) is the highest derivative that appears in the equation.
Linear Differential Equations - A differential equation is linear if the dependant variable and all of its derivatives appear in a linear fashion \(\frac{ dy }{ dx } + y =x\). They can not be multiplied together or squared for example or they are not part of transcendental functions such as sins, cosines, exponentials, etc., such as \(\;\frac{ dy }{ dx } + y^2 =x\).
Difference Between Terms
- \(\large{ \Delta }\) (Greek) = Meaning: finite, perceptible change - Usage: to find the difference.
- means a change in some variable. Operation: \( \Delta = initial - final \) or \({\large \frac{ \Delta y }{ \Delta x } = \frac{ y_i \;-\; y_f }{ x_i \;-\; x_f} }\)
- \(\large{ \delta }\) (Greek) = Meaning: approximation, non exact functions, estimates - Usage: to find an approximation.
- means a "small" or "infinitesimal" change like d, but when you don't want to talk about differential forms or derivatives using the other notations.
- \(\large{ d }\) (Latin) = Meaning: infinitesimal, imperceptible change - Usage: to differentiate.
- means an "infinitesimal" change, or the "total derivative" (exact differential). It's kind of like a limit as \( \Delta v -> 0 \), but it is compatible with relative rates or different kinds of limits, so that the notion of a derivative is preserved (in the form dy/dx, for example.)
- \(\large{ \partial }\) (Cryillic) = Meaning: infinitesimal change - Usage: partial differentiate.
- means a "partial" differential. It's basically the same as d, except it also tells you that there are other related variables that are being held constant. In other words, it is never a complete picture. It's typically used in partial derivatives, derivatives that are only in one dimension of a larger dimensional space.
- \(\large{ \delta }\), \(\large{ d }\), and \(\large{ \partial }\) are not used interchangeably.
- In summary, the choice of symbol provides context about the type and magnitude of the change being described: for large finite differences ( \(\large{ \Delta}\) ) to infinitesimals in single-variable ( \(\large{ d}\) ) vs. multi-variable ( \(\large{\partial}\) ) contexts, and specific variations in physics ( \(\large{ \delta}\) ).
Derivative Rules |
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| Rules |
Function | Derivative |
| Constant Function Rule | \(\large{ \frac{d}{dx}\;a = 0 }\) (where a = any constant) | |
| Scalar Multiple Rule | \(\large{ \frac{d}{dx}\left(au \right) = a\;\frac{du}{dx} }\) (where a = any constant) | |
| Sum Rule | \(\large{ \frac{d}{dx}\left(u+v \right) = \frac{du}{dx} + \frac{dv}{dx} }\) | |
| Difference Rule | \(\large{ \frac{d}{dx}\left(u-v \right) = \frac{du}{dx} - \frac{dv}{dx} }\) | |
| Power Rule | \(\large{ \frac{d}{dx} u^n = nu^{n-1} \frac{du}{dx} }\) | |
| Product Rule | \(\large{ \frac{d}{dx}\left(uv \right) = u'v+uv' }\) | |
| Quotient Rule | \(\large{ \frac{d}{dx}\left(\frac{u}{v} \right) = \frac{u'v-uv'}{v^2} }\) | |
| Reciprocal Rule | \(\large{ \frac{d}{dx}\left(\frac{1}{v} \right) = \frac{v'}{v^2} }\) | |
| Chane Rule | \(\large{ \frac{d}{dx} f \left( g\left( x \right) \right) = f' \left( g\left( x \right) \right) g'\left( x \right) }\) or \(\large{ \frac{dy}{dx} = \frac{dy}{du}\;\frac{du}{dx} }\) | |
| Trig Function Rule |
\(\large{ \frac{d}{dx} \; sin\;u = cos\;u \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; cos\;u = sin\;u \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; tan\;u = sec^2\;u \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; cot\;u = csc^2\;u \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; sec\;u = sec\;u \;tan\;u \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; csc\;u = csc\;u \;cot\;u \;\frac{du}{dx} }\) |
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| Inverse Trig Function Rule |
\(\large{ \frac{d}{dx} \; sin^{-1}\;u = \frac{1}{\sqrt{1\;-\;u^2 } } \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; cos^{-1}\;u = -\; \frac{1}{\sqrt{1\;-\;u^2 } } \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; tan^{-1}\;u = \frac{1}{1\;+\;u^2 } \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; cot^{-1}\;u = -\; \frac{1}{1\;+\;u^2 } \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; sec^{-1}\;u = \frac{1}{\left\vert u \right\vert\;\sqrt{u^2\;-\; 1 } } \;\frac{du}{dx} }\) \(\large{ \frac{d}{dx} \; csc^{-1}\;u = -\; \frac{1}{\left\vert u \right\vert\;\sqrt{u^2\;-\; 1 } } \;\frac{du}{dx} }\) |
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| Expotential Function Rule | \(\large{ \frac{d}{dx} \left( \epsilon^{\alpha} \right) = \epsilon^{\alpha} \; \frac{du}{dx} }\) and \(\large{ \frac{d}{dx} \left( a^{\alpha} \right) = a^{\alpha} \left( ln\;a \right) \; \frac{du}{dx} }\) | |
| Log Function Rule | \(\large{ \frac{d}{dx} \left( ln\;u \right) = \frac{1}{u} \; \frac{du}{dx} }\) and \(\large{ \frac{d}{dx} \left( log_{\lambda} \;u \right) = \frac{1}{ \left( ln\;a \right) u } \; \frac{du}{dx} }\) | |
| Inverse Function Rule | \(\large{ \frac{d}{dx} f^{-1} \left( x \right) = \frac{ 1 }{ f' \left( f^{-1} \left( x \right) \right) } }\) | |
