Editing Slope (section)

==Calculus==
[[File:Tangent function animation.gif|right|frame|At each point, the [[derivative]] is the slope of a [[Line (geometry)|line]] that is [[tangent]] to the [[curve]] at that point. Note: the derivative at point A is [[positive number|positive]] where green and dash–dot, [[negative number|negative]] where red and dashed, and [[zero (number)|zero]] where black and solid.]]
The concept of a slope is central to [[differential calculus]]. For non-linear functions, the rate of change varies along the curve. The [[derivative]] of the function at a point is the slope of the line [[tangent]] to the curve at the point and is thus equal to the rate of change of the function at that point.

If we let Δ''x'' and Δ''y'' be the distances (along the ''x'' and ''y'' axes, respectively) between two points on a curve, then the slope given by the above definition,
:<math>m = \frac{\Delta y}{\Delta x}</math>,

is the slope of a [[secant line]] to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve.

For example, the slope of the secant intersecting ''y'' = ''x''<sup>2</sup> at (0,0) and (3,9) is 3. (The slope of the tangent at {{nowrap|1=''x'' = {{frac|3|2}}}} is also 3&nbsp;−&nbsp;''a'' consequence of the [[mean value theorem]].)

By moving the two points closer together so that Δ''y'' and Δ''x'' decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using [[differential calculus]], we can determine the [[limit of a function|limit]], or the value that Δ''y''/Δ''x'' approaches as Δ''y'' and Δ''x'' get closer to [[zero]]; it follows that this limit is the exact slope of the tangent. If ''y'' is dependent on ''x'', then it is sufficient to take the limit where only Δ''x'' approaches zero. Therefore, the slope of the tangent is the limit of Δ''y''/Δ''x'' as Δ''x'' approaches zero, or d''y''/d''x''. We call this limit the [[derivative (calculus)|derivative]].

:<math>\frac{\mathrm dy}{\mathrm dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}</math>

The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, let ''y'' = ''x''<sup>2</sup>. A point on this function is (−2,4). The derivative of this function is {{nowrap|1={{frac|d''y''|d''x''}} = 2''x''}}. So the slope of the line tangent to ''y'' at (−2,4) is {{nowrap|1=2 ⋅ (−2) = −4}}. The equation of this tangent line is: {{nowrap|1=''y'' − 4 = (−4)(''x'' − (−2))}} or {{nowrap|1=''y'' = −4''x'' − 4}}.