Changes

:<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#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.

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>