Changes

A '''derivative''', one of the fundamental concepts of [[calculus]], measures how quickly a [[function]] changes as its input value changes.  Given a graph of a [[real]] curve, the derivative at a specific point will equal the [[slope]] of the line [[tangent]] to that point.  For example, the derivative of <tt>y = x²</tt> at the point <tt>(1,1)</tt> tells how quickly the function is increasing at that point.  If a function has a derivative at some point, it is said to be '''differentiable''' there.  If a function has a derivative at every point where it is defined, we say it is a '''differentiable function'''.  Differentiability implies [[continuous|continuity]]  as well as [[integral|integrability]] on bounded [[subset]]s of the [[domain]].

A '''derivative''', one of the fundamental concepts of [[calculus]], measures how quickly a [[function]] changes as its input value changes.  Given a graph of a [[real]] curve, the derivative at a specific point will equal the [[slope]] of the line [[tangent]] to that point.  For example, the derivative of <tt>y = x²</tt> at the point <tt>(1,1)</tt> tells how quickly the function is increasing at that point.  If a function has a derivative at some point, it is said to be '''differentiable''' there.  If a function has a derivative at every point where it is defined, we say it is a '''differentiable function'''.  Differentiability implies [[continuous|continuity]].

One of the main applications of differential calculus is '''differentiating''' a function, or calculating its derivative.  The First [[Fundamental Theorem of Calculus]] explains that one can find the original function, given its derivative, by [[integration|integrating]], or taking the integral of, the derivative.

One of the main applications of differential calculus is '''differentiating''' a function, or calculating its derivative.  The First [[Fundamental Theorem of Calculus]] explains that one can find the original function, given its derivative, by [[integration|integrating]], or taking the integral of, the derivative.

==Definition==

==Definition==

The derivative of the function <tt>f(x)</tt>, <tt>f'(x)</tt> or :<math>frac{df}{dx}</math>, is defined as:

The derivative of the function <tt>f(x)</tt>, <tt>f'(x)</tt> or <math>\frac{df}{dx}</math>, is defined as:

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

:<math>f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}</math>

In other words, it is the limit of the [[slope]] of the [[secant]] line to f(x) as it becomes a [[tangent line]].  If the tangent line is increasing (which it is if the original function is increasing), the derivative is positive; if the function is decreasing, the derivative is negative.

In other words, it is the limit of the [[slope]] of the [[secant]] line to f(x) as it becomes a [[tangent#Other_Uses|tangent line]].  If the tangent line is increasing (which it is if the original function is increasing), the derivative is positive; if the function is decreasing, the derivative is negative.

For example, :<math>\frac{d}{dx} ( 2x ) =\lim_{h \to 0}\frac{2(x+h)-2(x)}{h} = \lim_{h \to 0}\frac{2x+2h-2x}{h} = \lim_{h \to 0}\frac{2h}{h} = \lim_{h \to 0}\ 2 = 2</math>

For example, :<math>\frac{d}{dx} ( 2x ) =\lim_{h \to 0}\frac{2(x+h)-2(x)}{h} = \lim_{h \to 0}\frac{2x+2h-2x}{h} = \lim_{h \to 0}\frac{2h}{h} = \lim_{h \to 0}\ 2 = 2</math>