**Differentiation and The Derivative**

**Differentiation** is a method to compute the rate at which a dependent output *y* changes with respect to the change in the independent input *x*. This rate of change is called the **derivative** of *y* with respect to *x*. In more precise language, the dependence of *y* upon *x* means that *y* is a function of *x*. This functional relationship is often denoted *y* = *f*(*x*), where *f* denotes the function. If *x* and *y* are real numbers, and if the graph of *y* is plotted against *x*, the derivative measures the slope of this graph at each point.

The simplest case is when *y* is a linear function of *x*, meaning that the graph of *y* divided by *x* is a straight line. In this case, *y* = *f*(*x*) = *m* *x* + *b*, for real numbers *m* and *b*, and the slope *m* is given by

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because

*y*+ Δ*y*=*f*(*x*+ Δ*x*) =*m*(*x*+ Δ*x*) +*b*=*m**x*+*b*+*m*Δ*x*=*y*+*m*Δ*x*.

It follows that Δ*y* = *m* Δ*x*.

This gives an exact value for the slope of a straight line. If the function *f* is not linear (i.e. its graph is not a straight line), however, then the change in *y* divided by the change in *x* varies: differentiation is a method to find an exact value for this rate of change at any given value of *x*.

**Figure 2.**The secant to curve

*y*=

*f*(

*x*) determined by points (

*x*,

*f*(

*x*)) and (

*x*+

*h*,

*f*(

*x*+

*h*))

**Figure 3.**The tangent line as limit of secants

The idea, illustrated by Figures 1-3, is to compute the rate of change as the limiting value of the ratio of the differences Δ*y* / Δ*x* as Δ*x* becomes infinitely small.

In Leibniz's notation, such an infinitesimal change in *x* is denoted by *dx*, and the derivative of *y* with respect to *x* is written

suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of *y* with respect to *x*", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

The most common approach to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis.

Read more about this topic: Derivative

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