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Common derivatives are frequently seen and easily calculated, while complex derivatives are rare and difficult. They are used in calculating rate of change, finding maxima and minima, and predicting behavior. Common integrals are related to common derivatives. They are often used in calculus and can be found in textbooks and online. Real-world application can be difficult, but drawing diagrams or identifying the function can help.
In calculus, a derivative is a measure of the rate of change of a mathematical function. The term “common derivative” simply refers to a type of derivative that is frequently seen or can be evaluated with relative ease. Conversely, complex derivatives are relatively rare and can be difficult to calculate.
Most of the derivatives found in most mathematical applications are common derivatives. For example, polynomials are functions composed of everyday mathematical operators on a variable; some examples are 3x, x4, and 2×2 + 5x + 12. These are all polynomials because they are all functions using the most commonly used mathematical operators on x. Consequently, the derivatives of these and other similar functions are considered common derivatives. Not only are the most basic rules of derivation used in their calculations, but more importantly, these functions are the most likely types to encounter.
When derived, the most commonly used mathematical functions give rise to common derivatives. Derivatives for trigonometric functions are often seen and calculated relatively quickly. Other functions with derivatives that can be described as common are logarithms and functions that raise a number to a positive exponent.
There is a close relationship between common derivatives and common integrals. In much the same way that an integral is simply an antiderivative, common integrals are just common antiderivatives. Graphs of common derivatives and integrals are usually found in most calculus textbooks and are available online.
Common derivatives find application as the basis for most mathematical calculations involving rate of change. Velocity is probably the best-known type of calculation for a rate of change. It is simply a derivative of position with respect to time; when an object is in motion, the rate of change of distance from another stationary or moving object can be calculated using a common derivative. A common derivative can also be useful in determining the relative maxima or minima of a function, which can help predict behaviors for any object related to that function.
Although many people who study mathematics become proficient in the calculus of common derivatives, real-world application tends to be more difficult. In such circumstances, it is sometimes useful to determine which function could be causing the described behavior. Another potentially useful way to approach the problem is to draw a simple diagram of the situation represented. Both of these methods can betray the information needed to arrive at a solution.
Derivatives are usually the first major new concept introduced to a calculus student. Common derivatives are simple enough in concept that many formulas exist for their solutions. Despite this, they remain one of the more obscure but useful concepts in mathematics.
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