Title 19. Differentiation Medium.pdf ✅

1.). f(2) = 4, f(2) =1, then = − − → x 2 xf(2) 2f(x) lim x 2 (a.) 1 (b.) 2 (c.) 3 (d.) -2 2.).If f(x + y) = f(x).f(y) for all x and y and f(5) = 2, f (0) = 3, then f(5) will be (a.) 2 (b.) 4 (c.) 6 (d.) 8 3.).If f(a) = 3,f(a) = −2,g(a) = −1,g(a) = 4, then x a g x f a g a f x x a − − → ( ) ( ) ( ) ( ) lim = (a.) – 5 (b.) 10 (c.) – 10 (d.) 5 4.).If x 2 x 1 5f(x) 3f  = +      + and y = xf (x) then x 1 dx dy =       is equal to (a.) 14 (b.) 8 7 (c.) 1 (d.) None of these 5.).The derivative of 3 f(x) =| x | at x = 0 is (a.) 0 (b.) 1 (c.) –1 (d.) Not defined 6.).The first derivative of the function ( sin2 cos2 cos3 log 2 ) 3 2 + + x x x x with respect to x at x =  is (a.) 2 (b.) –1 (c.) 2 2 loge 2  − + (d.) −2 + loge 2 7.).If y =| cos x | +| sin x | then dx dy at 3 2 x = is (a.) 2 1 − 3 (b.) 0
(c.) ( 3 1) 2 1 − (d.) None of these 8.).If f(x) log (log x), = x then f (x) at x = e is (a.) e (b.) 1/e (c.) 1 (d.) None of these 9.).If f (x) =| log x |, then for x  1, f (x) equals (a.) x 1 (b.) | | 1 x (c.) x −1 (d.) None of these 10.).If                 − = − 2 2 1 exp tan x y x x then dx dy equals (a.) 2 [1 tan (log )] sec (log ) 2 x + x + x x (b.) [1 tan (log )] sec (log ) 2 x + x + x (c.) 2 [1 tan (log )] sec (log ) 2 2 x + x + x x (d.) 2 [1 tan (log )] sec (log ) 2 x + x + x 11.).If , 2 ( ) cot 1         − = − − x x x x f x then f'(1) is equal to (a.) – 1 (b.) 1 (c.) log 2 (d.) −log 2 12.).If (1 )(1 )(1 ).......( 1 ) 2 4 2 n y = + x + x + x + x then dx dy at x = 0 is (a.) 1 (b.) – 1 (c.) 0 (d.) None of these 13.).If f(x) = cosx.cos2x.cos4x.cos8x.cos16x then        4  f is
(a.) 2 (b.) 2 1 (c.) 0 (d.) None of these 14.).If xe y x xy 2 = + sin , then at x = 0 , dx dy = (a.) – 1 (b.) – 2 (c.) 1 (d.) 2 15.).If sin( x + y) = log( x + y) , then dx dy = (a.) 2 (b.) – 2 (c.) 1 (d.) – 1 16.).If ln(x + y) = 2xy, then y (0) = (a.) 1 (b.) – 1 (c.) 2 (d.) 0 17.).If 2( ) , m n m n x y x y + = + the value of dx dy is (a.) x + y (b.) y x (c.) x y (d.) x − y 18.). If x = a(cos + sin) , = − = dx dy y a(sin  cos), (a.) cos (b.) tan (c.) sec (d.) cosec 19.).If 2 1 1 cos t x + = and 2 1 sin t t y + = , then dx dy = (a.) – 1 (b.) 2 1 1 t t + −
(c.) 2 1 1 + t (d.) 1 20.). If 2 2 1 t 1 t x + − = and 2 1 t 2t y + = , then = dx dy (a.) x −y (b.) x y (c.) y −x (d.) y x 21.).The first derivative of the function         +        −  + x x x 2 1 cos sin 1 with respect to x at x =1 is (a.) 4 3 (b.) 0 (c.) 2 1 (d.) 2 1 − 22.).If y x 1 x , n 2       = + + then dx dy x dx d y (1 x ) 2 2 2 + + is (a.) n y 2 (b.) n y 2 − (c.) −y (d.) x y 2 2 23.).If ( ) , n f x = x then the value of ! ( 1) (1) ...... 3 ! ' ' '(1) 2! ' '(1) 1! '(1) (1) n f f f f f n n − − + − + + is (a.) n 2 (b.) 1 2 n− (c.) 0 (d.) 1 24.).If         − + +                     = − − 1 6log x 3 2log x tan log(ex ) x e log f(x) tan 1 2 2 1 , then n n dx d y is (n  1) (a.) tan {(log ) } 1 n x − (b.) 0 (c.) 1/2 (d.) None of these