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ASSIGNMENT CLASS XII DIFFERENTIATION 1. Find
dy dx
for the following: 1
(a) y
a2 x2
1 x
4. If x
5. Find
dy dx
2
a2 2
1
dy dx
, prove that ddxy
ny 2
.
1
cos sin x cos x sin x
1 cos sin x
(c) tan 1
(b) tan
1 sin 1 sin x 1 s i n 1 s i n x
(e) tan 1
1 2 1 x 2 (d) tan 1 x 2 1 x 2 2
5 12 1 x 2 (e) sin 13 1
dy
1 y a x y , prove that
8. If y x
2
1
x 1 x, show that
2
10. If log x x y , prove that
dy dx
dx
dy , prove that 2 x y 2 dx
x 1 lo log
1 x 2 1 (c) tan x
1 (b) cot 1 x 1
3
1
2
a
2
(a) cos ( 4 x 3 x)
9. If
1 sin 2
for the following:
1
7. If 1
1 sin 2 x
for the following:
1 sin 1 sin
dx
cos sin x
y 0 .
n
(d) tan 1
dy
(f) y
cos sin x
x 2 2 a x . a
(a) sin 1 (cos ) cos 1 (sin x)
6. Find
(c) y
sin
, prove that (1 x 2 )
x2 a2
1 x 2
sin 2 ( 2 x 3)
(e) y log( x 1 x2 )
a x 2
dx 2
1
3. If y
2
1 e2 x
d x
2. Show that
3
1 cos
(d) y log
5 x
(b) y
1 y 2 1 x2
.
x.
2
x1
log x
1 log
2
.
dy dx
x1y 0 .
1
11. If log
x 2 y 2 tan 1
sin 1 x
12. If y
1
2
, prove that 1
13. If x cos x cos x sin x , find
14. If xa yb x y
a b
3 x 15. If f ( x ) 1 x
dy
, prove that
dy dx
2
dy
x
dx
dx
x y x y
.
xy1 .
.
, prove that
dy
dx
y
.
x
2 3 x
, find f ' 0 .
2 x 1 2 x . w . r . t . si s i n 2 2 1 1 x x
16. Differentiate tan 1
b dy . dx att a
17. If x a sin 2 t 1 cos 2 t , y b cos 2 t 1 cos 2 t , show that
4
1 t 2 dy 1 t 2 2t 18. If x a , show that . , y 2 2 1 1 t t d x 2 a t d2y 19. If x 2 cos cos 2 and y 2 sin sin 2 , find . 2 dx 2
20. If Acos nx Bsin nx, prove that
21. If e sin x cos x , prove that x
22. If tan
1
x , show that 1 x
2
d2y dx
d2y dx 2
d2y
dx
2
2
n2 y 0 .
2
2x
dy dx
dy dx
2 y 0 .
0.
ANSWERS 1 .(a)
(a x ) 2
(f) sec 2 ( (d) 15 .
15 5 x 2
x 2 32
4
1 x4
sin xcos x e2 x (b) 2 sin( 4 x 6) (c) sec ( x ) (d) 2 43 2 2 x 4 3(1 x ) 1 cos 1 e
x ) (e)
27 lo log 3 12
5.
(a) 2
1 1
2
2
16.
13 .
1
cos x
(b)
1 2
(c) 1 (d)
1 2
(e)
1 2
6.(a)
3 1 x 2
(b)
1 1 x
2
1
(e) (c)
1 x 2 1 2(1
cos x log x sin x cos xsin x sin xtan x cos xlog cos x x 19 .
