Question

Prove the formula for (d/dx)(cos−1(x)) by the same method as for (d/dx)(sin−1(x)). Let y = cos−1(x). Then cos(y) = and 0 ≤ y ≤ π ⇒ −sin(y) dy dx = 1 ⇒ dy dx = − 1 sin(y) = − 1 1 − cos2 = − 1 1 − x2 .

Answer 1

`(d\,arccos(x))/(dx) =-(1)/(√(1-x^2) )`

See the proof below

Explanation:

if
`y=arccos(x)`, then :
`cos(y)=cos(arccos(x))=x`

We can then apply the derivative to both sides of this last equation and remember to use the chain rule as we encounter the variable "y":

`cos(y)=x\\(d)/(dx) cos(y)=(d)/(dx) x\\-sin(y)\,(dy)/(dx) =1\\(dy)/(dx) =-(1)/(sin(y)) \\(dy)/(dx) =-(1)/(√(1-cos^2(y)) )\\(dy)/(dx) =-(1)/(√(1-x^2) )\\(d\,arccos(x))/(dx) =-(1)/(√(1-x^2) )`