Question

differentiation + integration

Answer 1

Suppose\[0\le x \le 1/2\]
(a) Using implicit differentiation, show that \[d/dx (\cos^{-1} x) = -1/(\sqrt{1-x ^{2}})\]

Answer 2

(b) Using integration by parts, find \[\int\limits_{0}^{-1/2} (\cos^{-1} x)/(\sqrt{1-x ^{2}}) dx\]

Answer 3

(a) proving?

Answer 4

yes

Answer 5

Let y = arccos x. So, x = cosy. draw a right triangle that denotes this equality...
solving for the missing side of the triangle you should get sqrt(1-x^2).
now implicitly differentiate for y':
[x = cosy]'
1 = -siny * y'
so y' = 1/(-siny). But according to the triangle siny=sqrt(1-x^2).
so y' = -1/sqrt(1-x^2).

thats the differentiation part.