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Application of Derivative: Verify Lines are Parallel

 Post to:   # Slope The slope of a line in the plane is defined as the rise over the run, m = Δyx.

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.

The slope is (in the simplest of terms) the measurement of a line, and is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Given two points (x1,y1) and (x2,y2) on a line, the slope m of the line is Through differential calculus, one can calculate the slope of the tangent line to a curve at a point.

The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the grade m of a road is related to its angle of incline θ by http://en.wikipedia.org/wiki/Slope

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity (conversely, integrating a car's velocity over time yields the distance traveled).

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.

The process of finding a derivative is called differentiation.

## Computing the derivative

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.

### Derivatives of elementary functions

Most derivative computations eventually require taking the derivative of some common functions. The following incomplete list gives some of the most frequently used functions of a single real variable and their derivatives.

Derivatives of powers: if where r is any real number, then wherever this function is defined. For example, if f(x) = x1 / 4, then and the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zero for x ≠ 0, which is almost the constant rule.

Constant rule:if ƒ(x) is constant, then (Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Derivative