d(cos^-1 sqrtcosx)/dxdcos1cosxdx equals ??

1 Answer
Mar 7, 2018

(tan(x)sqrt(cos(x)))/(2(sqrt(1-cos(x)))tan(x)cos(x)2(1cos(x))

Explanation:

First let's apply u substitution making uu =sqrt(cos(x))cos(x) Making d/dxarccos(u)ddxarccos(u).

We must note d/dx arccos(x) = (-1)/(1-x^2)ddxarccos(x)=11x2. Applying this to arccos(sqrt(cos(x)))arccos(cos(x)) we receive d/dx arccos(x) = (-1)/(1-(u)^2)ddxarccos(x)=11(u)2. Applying the chain rule we then multiple the function by u' leaving d/dx arccos(x) = (-u')/sqrt((1-(u)^2)).
u' = (-sin(x))/(2(sqrt(cos(x))))
Now me must substitute u into the function d/dx arccos(x) = (-(-sin(x))/(2(sqrt(cos(x)))))/sqrt((1-(sqrt(cos(x)))^2)

We must further simplify this function (1-(sqrt(cos(x)))^2) = 1-cos(x)

(-sin(x))/(2(sqrt(cos(x)))) rationalizing the denominator we recieve
(sqrt(cos(x)))(-sin(x))/(2(cos(x)) or since sin(x)/(cos(x)) = tan(x) then (-tan(x))(sqrt(cos(x)))/2

furthermore we remove the negative in front of tan(x) from the -u leaving (tan(x)sqrt(cos(x)))/(2(sqrt(1-cos(x)))