## Vineet Loomba is teaching live on Unacademy Plus

IIT-JEE VIDEO COURSE FOR SURE SHOT SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) IIT-JEE MATHS MADE EASY PREPARED BY: ER. VINEET LOOMBA IITian | IIT-JEE MENtOR Search vineet loomba unacademy" on GOOGLE

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) ABOUT ME B.Tech. From IIT Roorkee IIT-JEE Mentor Since 2010 Doubts/Feedback in Comment Section * Comment other topics you want to revise. Follow me @ https://unacademy.com/user/vineetloomba to get updates or search me on Google * Share among your peers as SHARING is CARING!!

DIFFERENTIABILITY IIT-IEE MATHS MADE EASY PREPARED BY: ER. VINEET LOOMBA IITiAN | IIT-JEE MENTOR FoR SURE SHoT SUcCESS IN JEE MAIN AND ADVANCED (IIT-JEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Let f(x) be a real valued function defined on an open interval (a, b) and let c E (a, b). Then fx) is said to be differentiable on derivable at x -c. If lim This limit is called the derivative or differential cofficient of the function f(x) at x-c and is denoted by f"(c) or Df(c) or {fx) Thus f(x) is differentiable at x c f(x)-flc) exists finitely f(x)-f(c) f(x)-f(c) f(x)-f(c) lim exists finitely lim = lim X-C fc+h)-f(c)lm lim fc-h)-fc) MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKee)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) f(c-h)-f(c) -h Here lim is called left hand derivative and f(c +h)-f(c) h is dented by f(c) or Lf(c) while lim- is called right hand derivative and is denoted by f(c) or Rf (c) Thus f(x) is differentiable at x = c if Graphically : fe-hi-f(c) f(c-h)| flc) Qfc) Slope of chord PQ = -h c-h cc+ h Slope of chord QR- fc-fc+h) MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Tangent is always limiting position of chord So as h - 0, PQ and QR becomes tangents at P from left hand side and Right hand side lim f(c-h)-foc) tangent at P from left side f(c)-f(c+h) - tangent at P from Right side lim lf lim flc-h)-f c) then it implies that there is unique tangent at P A very important point to note that fx) is differentiable at a point P if the curve do not have P is as a comer point. imfc)-f cth) -h MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) DIFFERENTIABILITY IN AN INTERVAL : For open interval (a, b) A function fix) defined on an open interval (a, b) is said to be differentiable or derivable in open interval (a, b) if it is differentiable at each point of (a, b) For close interval [a, bl A function f(x) defined on [a, b] is said to be differentiable or derivable at the end points a and b if it is differentiable from the right at a and from the left at b. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) In other words f(x)-f(a and d lmf)-f(b) both exist. and -b im f(x)-f(a) If f is derivable in the open interval (a, b) and also at end points a and b, then f is said to be derivable in the closed interval [a, bl A function f is said to be a differentiable function if it is differentiable at every point of its domain If a function is differentiable at each x E R, then it is said to be everywhere differentiable. For example, a constant function, a polynomial function, sinx, cosx etc. are everywhere differentiable. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) SOME STANDARD RESULTS ON DIFFERENTIABILITY Every polynomial function is diferentiable at each x E R. the exponential function a, a 0 is differentiable at each x E R. Every constant function is differentiable at each x E R. The logarithmic function is differentiable at each point in its domain. Trigonometric and inverse-trigonometric functions are differentiable in their domains The sum, difference, product and quotient of two differentiable function is differentiable. The composition of differentiable function is a differentiable function. MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Show that f(x) = Ixl is not differentiable at x We have (LHD at x = 0) 0. Ex. Sol. lim fx)-f(o) f0-h)-f(O) lim f(-h)-f(0) =lim h20 0-h-0 [h1-1 - 1-hl = lim =lim-1=- = lim = lim and (RHD at x=0) f(o+h)-lim f(h)-f(O) lim f(x)-fro) h0h [h1-101 , t. lim 1-1 (LHD at x-O) # (RHD at x-o) lim h 0 h 0 So, f(x) is not differentiable at x = 0 ER. VINEET LOOMBA (IIT RooRKee) MATHEMATICS FOR IIT-JEE

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) Now, (LHD at x = 0) hsin, f(x)-f(0)_ f(0-h)-f(0) = lim = lim lim h 0 h 0 = (a number which oscillates between-1 and 1) (LHD at x 0) does not exist. Similarly, it can be shown that RHD at x = 0 does not exist. Hence, f(x) is not differentiable at x = o MATHEMATICS FOR IIT-JEE ER. VINEET LOOMBA (IIT RooRKEE)

FOR 100% SUCCESS IN JEE MAIN AND ADVANCED (IIT-JEE) ABOUT ME B.Tech. From IIT Roorkee IIT-JEE Mentor Since 2010 Doubts/Feedback in Comment Section * Comment other topics you want to revise. Follow me @ https://unacademy.com/user/vineetloomba to get updates. * Share among your peers as SHARING is CARING!!