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.
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.
Differential - A differential represents an infinitesimally small change in a quantity, used to describe how one variable changes with respect to another continuously. It is written as \(\;dx\), \(\;dy\), \(\;\dfrac{ dy }{ dx }\) for derivatives. The differential captures instantaneous rates of change rather than averages over intervals. For example, \(\;\dfrac{ dv }{ dl }\) is the differential form for acceleration, describing how velocity changes at a specific instant. Differentials are fundamental in physics, engineering, and mathematics for analyzing systems that vary smoothly and continuously, allowing precise modeling of motion, energy, and other changing quantities.
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 \( \bar{x} = \dfrac{ \sum x_i }{ n }\), where \(\;\sum x_i\) is the sum of all individual values and \(\;n\) 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.
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
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.
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 | \(\large{ \frac{d}{dx} f^{-1} \left( x \right) = \frac{ 1 }{ f' \left( f^{-1} \left( x \right) \right) } }\) | |
