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

Question

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

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

ans. y" = - cscθ cotθ

Answer 1

y = ln | csc θ - cot Θ | This is easier if we do a u substitution.

Let u = csc θ - cot Θ, Then y = ln | u |

du/dΘ = - csc Θ cot Θ - (-csc² Θ)

=. csc ² Θ - csc Θ cot Θ = csc Θ (csc Θ - cot Θ)

Since y = ln | u |

Then dy/du = 1/u

y' = dy/dΘ = (dy/du) (du/dΘ)

y' = (1/u) ( csc ² Θ - csc Θ cot Θ)

y' = (1/csc θ - cot Θ) ( csc Θ - cot Θ) The ( csc Θ - cot Θ) cancels. So...

y' = csc Θ

Then y'' = - csc Θ cot Θ

This problem can be done without the u substitution but the chain rule gets a little complex.