provided this limit exists Derivative of fat c is denoted by f c or dx function defined by wherever the limit exists is defined to be the derivative off The derivative of fis d dy dx dx denoted by f x or f x or if y f x by or y The process of finding derivative of a function is called differentiation We also use the phrase differentiate f x with respect to x to mean find f x The following rules were established as a part of algebra of deriva 1 ut v u v 2 uv u v uv Leibnitz or product rule 3 U lim h 0 120 u v uv its answer If lim h 0 wherever v 0 Quotient rule The following table gives a list of derivatives of rtain standard functions Table 5 3 f x f x tan x sin x sec x Whenever we defined derivative we had put a caution provided the limit exists Now the natural question is what if it doesn t The question is quite pertinent and so is f c h f c h does not exist we say that fis not differentiable at c In other words we say that a function fis differentiable at a point c in its domain if both f c h f c f c h f c h are finite and equal A function is said h to be differentiable in an interval a b if it is differentiable at every point of a b As in case of continuity at the end points a and b we take the right hand limit and left hand limit which are nothing but left hand derivative and right hand derivative of the function at a and b respectively Similarly a function is said to be differentiable in an interval a b if it is differentiable at every point of a b f x lim h 0 MATHEMATICS Therefore But for x c we have nx 1 and lim h 0 f x h f x h lim X C sin x Rationalised 2023 24 COS X f x f c Theorem 3 If a function f is differentiable at a point c then it is also continuous at that point Proof Since fis differentiable at c we have f x f c X C blish lim f x f c lim X C X C COS X f x f c X C f c 5 co J x c The x c f x f c X C 7 x c ed