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First Principles of Derivatives

As we noticed in the geometrical interpretation of differentiation, we can find the derivative of a function at a given point. If the derivative exists for every point of the function, then it is defined as the derivative of the function f(x).

Suppose f(x) is a real valued function, the function defined by 

lim{h right 0}{{f(x+h)~-~f(x)}/{h}}

For this function, if the derivative exists at every point along the curve, then we say 

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

This definition of derivative of f(x) is called the First Principle of Derivatives.

The function f'(x) or {dy}/{dx} is called the gradient function.

The process of finding the gradient value of a function at any point on the curve is called differentiation, and the gradient function is called the derivative of f(x).

There are different ways of representing the derivative of a function:

{dy}/{dx},  {df(x)}/{dx},   f'(x),   y’,   {delta y}/{delta x}, and {y over .}


Example 1: Find the derivative of f(x) = 5x using first principles.

f(x + h)  =  5(x + h)

{f(x+h)~-~f(x)}/{h}  =  {5(x+h)~-~5x}/{h}

= {5x~+~5h~-~5x}/{h}

= 5

f'(x)  =  lim{h right 0}{{f(x+h)~-~f(x)}/{h}}   =  5

Therefore f(x) = 5x;  and f'(x) = 5.


Example 2: Find the derivative of f(x) = 5x2 + 2x.

f(x + h)  =  5(x + h)2 + 2(x + h)

              =  5(x2 + 2xh + h2) + 2x + 2h

              =  5x2 + 10xh + 5h2 + 2x + 2h

f(x + h) – f(x)  =  (5x2 + 10xh + 5h2 + 2x + 2h)  –  (5x2 + 2x)

                       =  10xh + 5h2 + 2h

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

        =   lim{h right 0}{{10xh~+~5h^2~+~2h}/{h}}

        =   lim{h right 0}{10x~+~5h~+~2}

        =  10x + 2


There are more examples of derivatives here.