12%20-%20Differentiation

12%20-%20Differentiation - PROBLEMS (1) 12 -...

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PROBLEMS 12 - DIFFERENTIATION Page 1 ( 1 ) If f ( x ) = x sin x 1 for x 0 and f ( 0 ) = 0, prove that f is continuous but not differentiable at 0. ( 2 ) Find derivatives of the following functions using the definition of a derivative. ( i ) 1 x 1 x + - ( ii ) x 3 a ( iii ) sin 2 x ( iv ) x sin x [ Ans: ( i ) 2 ) 1 x ( 2 + , ( ii ) 3 x 3 a log a, ( iii ) 2x cos 2 x ( iv ) x cos x + sin x ] ( 3 ) If f ( x ) = x 1 sin x 2 , x 0 and f ( 0 ) = 0, prove that f ’ ( 0 ) = 0. ( 4 ) If f ( x ) = e x - 1, x 0 and f ( x ) = l sin x l , x < 0, is f continuous at 0 ? Is it differentiable at 0 ? [ Ans: continuous, not differentiable ] Find derivatives with respect to x of the following functions: ( 5 ) x log x sin x 2 + ) x log ( x sin x x sin x log 2x x cos x log x : Ans 2 2 - ( 6 ) x log x e 3 [ ] ) 3 log x 1 ( 3 : Ans x + ( 7 ) x 2 3 x sin x [ Ans: 3 x ( log 3 . x 2 sin x + 2x sin x + x 2 cos x ) ] ( 8 ) ) x ( log n n a a log x 1 : Ans ( 9 ) x log e x ) x log ( x ) 1 x log x ( e : Ans 2 x - ( 10 ) log [ log ( log x ) ] ) x log ( log x log x 1 : Ans
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PROBLEMS 12 - DIFFERENTIATION Page 2 Find derivatives with respect to x of the following functions: ( 11 ) log ( x + 2 2 a x + ) + a x 1 : Ans 2 2 ( 12 ) x 1 x 1 + - + ) x 1 ( ) x 1 ( 1 : Ans 2 3 2 1 - - ( 13 ) sin [ log l cos ( e x + x 2 ) l ] [ Ans: - ( e x + 2x ) tan ( e x + x 2 ) cos [ log l cos ( e x + x 2 ) l ] ( 14 ) sin [ cos { sin ( e x + 1 ) l ] [ Ans: - e x cos [ cos ( sin ( e x + 1 ) ) ] . sin [ sin ( e x + 1 ) ] . cos ( e x + 1 ) ] ( 15 ) l l x sin log e [ Ans: cos x if sin x > 0, - cos x if sin x < 0 ] ( 16 ) x sin e 2 x 2 tan + ) x 3 tan 2 2x sin ( x 2 tan e : Ans ( 17 ) log l sin ( tan x 2 ) l [ Ans: 2x cot ( tan x 2 ) sec 2 x 2 ] ( 18 ) , 2x sin 1 - 0 < x < 2 π < < + < < π π π π = 4 x at able differenti not , 2 x 4 for x sin x cos , 4 x 0 for x cos x sin : Ans - - ( 19 ) a x sin 1 - , 0 < l x l < l a l < > 0 a for x a 1 0, a for x a 1 : Ans 2 2 2 2 - - -
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PROBLEMS 12 - DIFFERENTIATION Page 3 Find derivatives with respect to x of the following functions: ( 20 ) x 1 2x sin 2 1 - - , l x l < 1 < = 2 1 x at able differenti not , 2 1 x for , x 1 2 1 , 2 1 2 1 1, x for , x 1 2 : Ans 2 2 l l l l - - - - - ( 21 ) ) 3x 4x ( cos 3 1 - - < = 2 1 x for able differenti not 1 , 2 1 2 1 1, x for x 1 3 , 2 1 x for x 1 3 : Ans 2 2 l l l l - - - - - ( 22 ) 1 2 2x
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12%20-%20Differentiation - PROBLEMS (1) 12 -...

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