Derivative of y = (1/x) + sqrt(cos x)

  • Thread starter Michele Nunes
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In summary, the conversation is discussing the evaluation of the derivative of the function y = (1/x) + sqrt(cos x) at the point (pi/2, 2/pi). The attempted solution involves using the power rule and chain rule, but there is a potential issue with the domain of the derivative due to the presence of sqrt(cos x). This is because there are intervals where cos(x) is negative, resulting in an undefined value for the derivative.
  • #1
Michele Nunes
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Homework Statement


Evaluate the derivative of the function y = (1/x) + sqrt(cos x) at the given point, (pi/2, 2/pi)

Homework Equations

The Attempt at a Solution


I used power rule on (1/x) and chain rule on sqrt(cos x) but when I was simplifying, there is a sqrt(cos x) in one of the denominators, which then becomes undefined when I plug in pi/2, so I'm confused what I did wrong, I'm pretty sure I used the correct rules and I'm pretty sure I simplified correctly, so why is it coming out undefined
 
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  • #2
Take a look at the domain of the derivative you calculated.
 
  • #3
Lord Anoobis said:
Take a look at the domain of the derivative you calculated.
What's more, Michelle, take a look at the domain of ##y = \frac 1 x + \sqrt{\cos x}##. Consider the fact that there are intervals where cos(x) is negative, so its square root isn't real on those intervals.
 

1. What is the derivative of y = (1/x) + sqrt(cos x)?

The derivative of y = (1/x) + sqrt(cos x) is -(1/x^2) - (sin x)/(2√cos x).

2. How do you find the derivative of y = (1/x) + sqrt(cos x)?

To find the derivative of y = (1/x) + sqrt(cos x), you can use the quotient rule and the chain rule. First, differentiate the numerator and denominator separately, then apply the quotient rule. For the square root term, use the chain rule to differentiate the inside function.

3. Can this function be simplified before finding the derivative?

Yes, this function can be simplified before finding the derivative. You can use algebraic manipulation to combine the two terms and rewrite the function as y = (1 + x√cos x) / x. This may make it easier to find the derivative using the quotient and chain rules.

4. What is the domain of the derivative for this function?

The domain of the derivative for this function is all real numbers except x = 0. This is because the function y = 1/x is not defined at x = 0, and the derivative of this function will also not be defined at that point.

5. Is there a simpler form for the derivative of this function?

Yes, there is a simpler form for the derivative of this function. By manipulating the function and using trigonometric identities, the derivative can be written as y' = (x√cos x - sin x) / x^2√cos x. This form may be more useful for further calculations or analysis.

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