what is the derivative of x/tan(x) and find f'(pi/4)

f'(x)=(tanx-x*sec(x)^2)/(tanx)^2
That's the quotient rule for derivatives or as my teacher says:

Low,low, low D'High, minus High d'low all over low square that's the quotient rule. It's a song which helps. D'High or d'low means the derivative of the top or bottom.

Anyways: the answer choices are:
2,1/2,1+pi/2,pi/2-1, 1-pi/2

and i keep getting 1-pi/8. This question is from an AP test in 1985 so i don't think they made a mistake in these answers.

deriv = (1*tan x - x sec^2 x)/tan^2 x

so for x= pi/4

deriv = (1 - pi/4(2))/1
=1 - pi/2, which was one of the choices

1-pi/2 appears to be correct.

To find the derivative of f(x) = x/tan(x), we can use the quotient rule.

The quotient rule states that for a function f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative is given by:

f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2

Using this rule, we can find the derivative of f(x):
f'(x) = (tan(x) - x * sec^2(x)) / (tan(x))^2

Now, let's find f'(pi/4):
Substituting x = pi/4 into f'(x), we get:
f'(pi/4) = (tan(pi/4) - (pi/4) * sec^2(pi/4)) / (tan(pi/4))^2

Since tan(pi/4) = 1 and sec^2(pi/4) = 2, we can simplify the equation:
f'(pi/4) = (1 - (pi/4) * 2) / 1

Calculating further, we have:
f'(pi/4) = (1 - (pi/2)) / 1
f'(pi/4) = 1 - pi/2

Therefore, the derivative of f(x) = x/tan(x) is (1 - pi/2), and when evaluated at x = pi/4, f'(pi/4) is also equal to 1 - pi/2. This matches one of the answer choices.

To find the derivative of x/tan(x), you can use the quotient rule. The quotient rule states that for a function f(x) = g(x)/h(x), the derivative f'(x) is given by (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2.

In this case, g(x) is equal to x and h(x) is equal to tan(x). Therefore, g'(x) is equal to 1 and h'(x) is equal to sec^2(x). Plugging these values into the quotient rule formula, we get:

f'(x) = (tan(x) * 1 - x * sec^2(x)) / (tan(x))^2

Simplifying further, we have:

f'(x) = (tan(x) - x * sec^2(x)) / (tan(x))^2

Now, to find f'(pi/4), we substitute x = pi/4 into the derivative function:

f'(pi/4) = (tan(pi/4) - (pi/4) * sec^2(pi/4)) / (tan(pi/4))^2

Since tan(pi/4) = 1 and sec^2(pi/4) = 2, we can simplify the equation to:

f'(pi/4) = (1 - (pi/4) * 2) / 1

Multiplying (pi/4) by 2 gives (pi/2), so we have:

f'(pi/4) = (1 - pi/2) / 1

Simplifying further, we get:

f'(pi/4) = 1 - pi/2

Therefore, the correct answer is 1 - pi/2.