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Shodor > CSERD > Resources > Applications > Differentiation



What are Derivatives?

The derivative refers to the rate of change, or slope, of a function. Differentiation refers to the calculation of a derivative.

Derivatives are used throughout applied mathematics and science. The first derivative most students come across is speed, which is the rate of change of position with respect to time (60 miles per hour means that if you were to keep driving at the same speed for one hour your position would change by 60 miles).

The derivative of a function is the instantaneous rate of change of a function evaluated at each point. This is written as

\frac{d}{d x} y = lim_{h -> 0} \frac{y(x+h)-y(x)}{h}

How are Derivatives Calculated?

Analytic Calculation

While it is possible to verify many formulas for calculating derivatives using the above definition, most of the time derivatives are calculated using a look-up table. Since a table including every single derivative in existance would be a bit cumbersome, often only the most fundamental derivatives are listed, and one is required to apply a variety of useful rules to calculate more complicated derivatives.

Some common derivatives are:

  • $\frac{d}{d x} x^a = a x^{a-1}$
  • $\frac{d}{d x} sin(x) = -cos(x)$
  • $\frac{d}{d x} exp(x) = exp(x)$
  • $\frac{d}{d x} ln(x) = x^{-1}$

    Some of the more common rules regarding derivative calculation are:

  • $\frac{d}{d x} k f(x) = k \frac{d}{d x} f(x),$ k = constant
  • $\frac{d}{d x} g(x) f(x) = g(x) \frac{d}{d x} f(x) + f(x) \frac{d}{d x} g(x)$
  • $\frac{d}{d x} f(y) = \frac{d y}{d x} \frac{d}{d y} f(y)$

    Numerical Calculation

    Another method for calculating derivatives involves approximating the exact limit above with a difference: $\frac{d}{d x} y = \frac{y(x+h)-y(x)}{h}$, h > 0.

    More complicated methods involve assuming a known simple function, such as a polynomial, can be fit to data at a few points.

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