Derivative or Differential Coefficient of a Function.





Differential calculus or the concept of Derivative and Differential Coefficient was discovered by Isaac Newton (1642-1727) and Gottfried Wilhelm Leibnitz (1646-1716) in the process of solving two old problems one of finding slope of tangent drawn to a curve and another of finding instantaneous velocity of an object in non-uniform motion.

Derivative:

When a variable “y” is defined as a function of another variable “x” or,

f(x)=y

Then , The Derivative or Differential Coefficient of the function “f” at a point “x” or with respect to “x” is the limiting value of:

\displaystyle\lim_{x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}

The derivative of a function of “x” with respect to “x” is denoted by:

\dfrac{df(x)}{d(x)}

for example:

If “y” is a function of “x”  or f(x)=y whose graph looks like:

Then the derivative of the function “f” with respect to “x” at point “x” is :-

\dfrac{df(x)}{d(x)}=\displaystyle\lim_{x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\displaystyle\lim_{x\to 0}\frac{\Delta y}{\Delta x}

which can be shown in figure as:



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