Find the indicated nth derivative of the following. Write the solution of each number

Question

7. 2nd derivative of y = In|cscθ - cotθ|

ans. y" = - cscθ cotθ

Answer 1

cscθ - cotθ = 1/sinθ - cosθ / sinθ = (1 - cosθ) / sinθ

So, y = ln l cscθ - cotθ l = ln l (1 - cosθ) / sinθ l = ln l 1 - cosθ l - ln l sinθ l

y' = (sinθ) / (1 - cosθ) - cosθ / sinθ = (sin²θ - cosθ + cos²θ) / [sinθ(1 - cosθ)] = (1 - cosθ) / [sinθ(1 - cosθ)]

So, y' = 1 / sinθ = (sinθ)^-1

Therefore, y" = -(sinθ)^-2(cosθ) = -cosθ / sin²θ = -(1 / sinθ)(cosθ / sinθ) = -cscθcotθ

Answer 2

y' = [1/(cscθ – cotθ)] × [d(cscθ)/dθ – d(cotθ)/dθ]

d(cscθ)/dθ = –cscθ cotθ and d(cotθ)/dθ = –csc²θ

which means that

y' = [1/(cscθ – cotθ)] × [–cscθ cotθ + csc²θ] = cscθ(cscθ – cotθ)/(cscθ – cotθ) = cscθ

y'' = d(y')/dθ = d(cscθ)dθ = –cscθ cotθ