Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ f x h f x f ' x lim h0 h x h1 x 1 x h1 x 1 lim h0 h 1 x 1 x h1 x 1 x h1 lim h0 h x 1 x h1 1 lim h0 h x 1 x h1 1 2h lim h0 h x 1 x h1 2 lim h0 x 1 x h1 2 2 2 2 x hx x x h1 x hx x x h1 x1 x1 x 1 2 Question 5: For the function 100 99 2 x x x F(x) = ... x 1 100 99 2 Prove that f’(1) = 100 f’(0) Solution 5: The given function is 100 99 2 x x x F(x) = ... x 1 100 99 2 100 99 2 d d x x x f ( x) ... x 1 dx dx 100 99 2 100 99 2 d d x d x d x d d f ( x) ... x 1 dx dx 100 dx 99 dx 2 dx dx d n n 1 On using theorem x nx ,we obtain dx 99 98 d 100x 99x 2x f( x) ... 1 0 dx 100 99 2 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 99 98 x x x ... 1 f x x x x 99 98 ' ... 1 At x =0, f ' 0 1 At x =1, 99 98 f f f ' 1 1 1 ... **11** **11** ... **11** **11**00 100 Thus, ' 1 100 ' 0 100terms Question 6: n n1 2 n2 n1 n Find the derivative of x ax a x ... a x a for some fixed real number a. Solution 6: n n1 2 n2 n1 n Let f(x) = x ax a x ... a x a d n n1 2 n2 n1 n f '( x) x ax a x ... a x a dx d n d n1 2 d n2 n1 d n d x a x a x ... a x a 1 dx dx dx dx dx d n n 1 On using theorem x nx ,we obtain dx n1 n2 2 n3 n1 n f '( x) nx a n 1 x a n 2 x ... a a 0 1 2 ... nx a n x a n x a n1 n2 2 n3 n1 Question 7: For some constants a and b, find the derivative of (i) (x â€“ a) (x â€“ b) (ii) (ax2 + b)2 (iii) x a xb Solution 7: (i) Let f (x) = (x â€“ a) (x â€“ b) Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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