`y = (1/x) + sqrt(cos(x)), ((pi/2),(2/pi))` Find and evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator


`y(x) = (1/x) + sqrt(cos(x))` .

cos(x) must be >=0 and cos(x) < 0 for `pigtxgtpi/2`  ` `(at the right neighborhood of `pi/2` ). So there can be only the left derivative.

Check that `y(pi/2) = 2/pi` :
`cos(pi/2) = 0` and `y(pi/2) = 2/pi + 0 = 2/pi` .

Next, find the derivative of y:
`y'(x) = -(1/x^2) + (1/2)*(cos(x))^(-1/2)*(-sin(x))` .

For `x-gtpi/2-0`

`y'(x) ->-oo`

(`1/x^2`  is finite, `sin(x)` is finite and `(cos(x))^(-1/2)-gt+oo` )

and therefore the tangent line is vertical, its equation is `x=pi/2` .

The graph is here: https://www.desmos.com/calculator/4owqqw7egh