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Geometrical Interpretation of Differentiation

Let y = f(x) be a function, and let P(a, f(a)) and Q(a+h, f(a+h)) be two points on the graph of the function that are close to each other. This graph is shown below:

geometrical representation of differentiation

Joining the points P and Q with a straight line gives us the secant on the graph of the function. And in the {Delta PQR} the gradient of the line is given by

m  =   {f(a+h)~-f(a)}/{a+h-a}  =   {f(a+h)~-f(a)}/{h}

In limiting process, i.e. as Q approaches P, h becomes really small, almost close to zero. So

m  =  {change in y}/{change in x}  

=   {Delta y}/{Delta x}  =   {f(a+h)~-~f(a)}/{h}

 lim{h right 0}~{{Delta y}/{Delta x}}  =  {f(a+h)~-~f(a)}/{h}

As {Q right P}, the chord/secant PQ tends to be a tangent at P for the curve y = f(x). Thus the limiting becomes the slope of the tangent at P for y = f(x).

We denote  lim{h right 0}~{{Delta y}/{Delta x}}  =   {dy}/{dx} , and

lim{h right 0}{{f(a+h)~-~f(a)}/{h}}  =  f'(a)

Therefore  {dy}/{dx}  =  f'(a)  =  m

m as we’ve seen is the slope or gradient of the tangent at P for f(x).

f'(a)  =  tantheta  = slope of f(x) at x = a.