Average
In formulas, the term average is the mean value of a quantity over a specific interval or set of data. It represents the total sum of all values divided by the number of values or the duration of time considered. For example, the average velocity is the total displacement divided by the total time taken, written as. 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.
In certain scientific and mathematical contexts, other types of averages are used depending on the nature of the data and what one wants to represent. The median (the middle value when data are ordered) is often preferred when distributions are skewed or contain outliers, as it better represents a typical value in such cases. The mode (the most frequently occurring value) is useful for categorical data or when identifying the most common outcome is important. Additionally, specialized averages appear in specific fields: the geometric mean is used for quantities that multiply together or grow exponentially (e.g., average growth rates, certain biological rates), calculated as the nth root of the product of n numbers. The harmonic mean appears in situations involving rates or ratios (e.g., average speed when distances are equal, or parallel resistances in physics), computed as the reciprocal of the arithmetic mean of the reciprocals.
Mean
The mean or mean value 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 http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mover><mi>x</mi><mo stretchy="false">¯</mo></mover></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mfrac><mrow><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mstyle></math>">, where http://www.w3.org/1998/Math/MathML"><mspace width="thickmathspace" /><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub></math>"> is the sum of all individual values and http://www.w3.org/1998/Math/MathML"><mspace width="thickmathspace" /><mi>n</mi></math>"> is the number of values. 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
In a formula, change is the difference between a final and an initial value of a quantity over a certain interval. It is represented using the Greek letter delta (Δ), such 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.
- Change in - The difference between final and initial values of a variable. Example:
In mathematics, a differential is the extremely small or infinitesimal change in a function associated with an infinitesimal change in its independent variable, formally defined asfor a function \(\;y = f(x)\;\). It represents the linear part of the change in the function, the amount the tangent line rises or falls when the input changes by the tiny quantity \(\;dx\;\) and serves as the foundation for linear approximations, the total differential in several variables \(\;(dz = fx \cdot dx + fy \cdot dy + ...)\;\), and the theory of differential forms in advanced geometry and topology. Differentials provide a precise language for handling rates of change and are central to the notation and solution methods of both ordinary and partial differential equations.
In science, particularly physics, engineering, and chemistry, the term differential almost always refers to an infinitesimal quantity of a physical variable ( such as \(\;dV\) for volume, \(\;ds\) for displacement, \(\;dW\) for work ) used when formulating continuous laws of nature. Basically, it is written as \(\;dx\), \(\;dy\), \(\;\dfrac{ dy }{ dx }\) for derivatives. These differentials allow scientists to express relationships that are true locally at every point in space and time, leading directly to the differential equations that govern most fundamental theories. While mathematics treats the differential as a rigorous object for approximation and integration, science uses it as the natural language for describing how physical quantities change continuously from one instant or point to the next.
- Differential is an infinitesimally small change in a variable. It represents how much a quantity changes when its input changes by an extremely tiny amount. In calculus, differentials are written as dx, dy, etc., and are used to describe derivatives, rates of change, and approximations.
- Differential refers to a very small change in a quantity. In mathematics and science, it is used to describe how one variable changes in relation to another when those changes are extremely tiny—almost infinitesimal. Differentials are the building blocks of calculus and help express rates of change, slopes of curves, and how systems respond to small variations.
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) or DIFF, 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.
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 an '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 a "differential form." It's kind of like a limit as \(\large{ \Delta v -> 0 }\), 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.
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 | \(\large{ \frac{d}{dx} f^{-1} \left( x \right) = \frac{ 1 }{ f' \left( f^{-1} \left( x \right) \right) } }\) | |
