Evaluate the derivative of the function below at the point (ð/3, 9.301). (Round to three decimal places.)
y= (9/x) + ((cosx)^.5)
y = 9/x + sqrt(cos x)
y' = -9/x^2 - .5sin x/sqrt(cos x)
y'(pi/3) = -9/(pi^2/9) - .5*sqrt(3)/2 / sqrt(.5)
= -81/pi^2 - sqrt(6)/4
= -8.819
To evaluate the derivative of the function y = (9/x) + (√cos(x)) at the point (ð/3, 9.301), we need to find the derivative of the function with respect to x and then substitute the x-coordinate of the given point (ð/3) into the derivative expression.
Step 1: Find the derivative of y with respect to x.
To find the derivative, we need to apply the rules of differentiation. Let's differentiate each term separately.
First term: 9/x
To differentiate 9/x, we can use the power rule for differentiation:
d/dx (9/x) = -9/x^2
Second term: √cos(x)
To differentiate √cos(x), we use the chain rule. Let's define u = cos(x).
Then, we have y = √u.
To differentiate y = √u with respect to x, we have:
dy/dx = (dy/du) * (du/dx)
Differentiating y = √u with respect to u gives:
dy/du = 1/(2√u)
Differentiating u = cos(x) with respect to x gives:
du/dx = -sin(x)
Now, applying the chain rule:
dy/dx = (dy/du) * (du/dx) = (1/(2√u)) * (-sin(x)) = -sin(x)/(2√u)
Step 2: Substitute the x-coordinate of the given point into the derivative expression.
We substitute x = ð/3 into the derivative expression we obtained in Step 1.
For the first term:
-9/(ð/3)^2 = -9/(ð^2/9) = -9*9/ð^2 = -81/ð^2
For the second term:
-sin(ð/3)/(2√cos(ð/3))
Calculate the value of sin(ð/3) and cos(ð/3) using the unit circle or a scientific calculator.
Substituting these values into the derivative expression, we get the value of the derivative at the point (ð/3, 9.301).