Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.Evaluate the derivative of the function below at the point (π/2, 2/π). (If an answer is...

Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

Evaluate the derivative of the function below at the point (π/2, 2/π). (If an answer is undefined, enter UNDEFINED. Round your answer to three decimal places.)

y=1/x +√cos x  (π/2, 2/π)

Asked on by ramiyaou

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the following limit to find derivative of the given function at `x = pi/2`  such that:

`f'(pi/2) = lim_(x->pi/2)(1/x + sqrt cos x - 2/pi)/(x - pi/2) `

Notice that substituting `pi/2`  for x yields an indetermination `0/0` , hence, you may use l'Hospital's theorem such that:

`lim_(x->pi/2)(1/x + sqrt cos x - 2/pi)/(x - pi/2) = lim_(x->pi/2)((1/x + sqrt cos x - 2/pi)')/((x - pi/2)')`

`lim_(x->pi/2)(1/x + sqrt cos x - 2/pi)/(x - pi/2) = lim_(x->pi/2)(-1/x^2 - sin x/(2sqrt(cos x)))/1`

`lim_(x->pi/2)(1/x + sqrt cos x - 2/pi)/(x - pi/2) = -4/(pi^2) - 1/0 = -oo`

Hence, evaluating the derivative of the given function at `x = pi/2`  yields `f'(pi/2) = -oo` , hence, you cannot find the derivative of the given function at `x = pi/2`  since it is undefined.

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