Question

find nth derivative of x=sin(logy/a)​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

x=sin (log y/a)

sin^-1 x= 1/a log y

a sin^-1 x= log y

e^(a sin^-1x) = y {take log on both sides}

differentiate following with respect to x

y1= a e^sin^-1x 1/√(1+x^2)

√(1+x^2)y1 = a y :{replace the valu of y}

squaring on both sides

(1+x^2)y1^2 = a^2y^2

again different following

2(1+x^2) y1 y2 +2xy1^2 = 2a^2yy1

devide the following 2y1 we get

(1+x^2) y2+ xy1 = a^2 y

and now to find nth derivative diffe following with libnize theorem

(1+x^2)yn+2 + 2nxy(n+1) +n(n-1) yn + yx(n+1) + nyn = a^2yn

(1+x^2)y(n+2) +(2n+1) yn+1 + (n^2-a^2)yn= 0