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How to Calculate the Flare Rate of an Exponential Horn

The term "exponential horn" is given to the shape of a graph generated by plotting an exponential function. A classic example of this is "Gabriel's Horn," a graph of the function f (x) = x^(-1). This specific exponential is notable for its infinite surface area and finite volume. The flare rate of a "horn" is the rate at which the curve changes with respect to the x-axis. This figure is calculated by taking the first derivative of the exponential being examined.

Instructions

- 1
  Place the equation into function notation. For example, the equation y = x^-2 becomes f (x) = x^-2.
- 2
  Take the first derivative of the function. Applying the general power rule of derivatives f (x) = x^-2 becomes f ' (x) = -2* (x^-3). Note the apostrophe in the notation; it means "F prime of x" and signifies the function is a derivative.
- 3
  Simplify the resulting derivative. In conclusion, f ' (x) = -2* (x^-3) is simplified as - (2 / x^3). This final function is the rate of change in the graph of the exponential function. It relates the rate at which the graph "flares" as it is followed along the x-axis.

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