What is the formal definition of a derivative?

1 Answer
May 29, 2015

The formal definition of derivative of a function #y=f(x)# is:
#y'=lim_(Deltax->0)(f(x+Deltax)-f(x))/(Deltax)#

The meaning of this is best understood observing the following diagram:

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The secant PQ represents the mean rate of change #(Deltay)/(Deltax)# of your function in the interval between #x# and #x+Deltax#.

If you want the rate of change, say, at P you "move" point Q (and the secant with it) to meet point P as in:
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In doing so you must reduce #Deltax#. If #Delta x->0# you'll get the tangent in P whose inclination will give the inclination at P and thus the derivative at P.

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Hope it helps!