Differential Calculus

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  Derivative

Derivative at a point : Let f(x) be a real valued  function defined  on open interval (a,b) and let c∈(a,b)  then f(x) is said to be differentiable  at x=c iff

\dpi{120} \mathbf{\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}}

exist finitely.

This limit is called  derivative of f(x) at x=c and is denoted by f ’(c) or   \dpi{120} \frac{\mathrm{d} (f(x))}{\mathrm{d} x}       so

\dpi{120} \mathbf{f'(c)= \lim_{h\rightarrow 0}\frac{f(c+h)-f(c)}{h}}

Derivative represents  slope of function. This is basically the change in y values (output values) with respect to change in x values (input values). It can also be represented  by   \dpi{120} \frac{\Delta y}{\Delta x}   or   \dpi{120} \frac{\mathrm{d} y}{\mathrm{d} x}  . It also represent  the average rate of change of function over an interval.

There are two ways   to find derivative.

  • Derivative using difference quotient or limit definition method.
  • Derivative using standard rules and formulas.

 

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