CONTINUITY AND DIFFERENTIABILITY
Example 41 If y = sin–1 x, show that (1 – x2)
d2y dy −x =0. 2 dx dx
Solution We have y = sin–1 x. Then
dy = dx
1 (1 − x 2 )
(1 − x 2 )
or
dy =1 dx
d ⎛ dy ⎞ 2 ⎜ (1 − x ) . ⎟ = 0 dx ⎝ dx ⎠
So
(
)
or
(1 − x 2 ) ⋅
d 2 y dy d + ⋅ dx 2 dx dx
or
(1 − x 2 ) ⋅
d 2 y dy 2x − ⋅ =0 2 dx 2 1 − x 2 dx
(1 − x 2 ) = 0
d2y dy −x =0 2 dx dx Alternatively, Given that y = sin–1 x, we have Hence
(1 − x 2 )
y1 =
So Hence
1 1− x
2
, i.e., (1 − x 2 ) y 2 = 1 1
(1 − x 2 ) . 2 y1 y2 + y12 (0 − 2 x) = 0 (1 – x2) y2 – xy1 = 0
EXERCISE 5.7 Find the second order derivatives of the functions given in Exercises 1 to 10. 1. x2 + 3x + 2 4. log x
2. x 20 3
5. x log x
3. x . cos x 6. ex sin 5x
7. e6x cos 3x 8. tan–1 x 9. log (log x) 10. sin (log x) d2y + y=0 11. If y = 5 cos x – 3 sin x, prove that dx 2
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