f(x) = cot x
f'(x) = -csc^2 x
The function f(x) = cot x is a trigonometric function defined as the ratio of the adjacent side to the opposite side in a right triangle. The cotangent function is the reciprocal of the tangent function (tan x) and is defined as:
f(x) = cos x / sin x
Therefore, in order to evaluate the function, we need to know the values of cosine and sine of x. We can determine these values using the unit circle or a calculator.
First, we need to identify any vertical asymptotes of the function. These occur when the denominator, sin x, equals zero. Thus, the vertical asymptotes occur at x = nπ, where n is any integer.
Next, we can evaluate the function at any other value of x. For example, if we want to find the value of f(x) at x = π/4, we can use the definitions of cosine and sine to get:
f(π/4) = cos(π/4) / sin(π/4)
= (√2 / 2) / (√2 / 2) = 1
Therefore, f(π/4) = 1.
Similarly, we can evaluate the function at other values of x. However, we should be careful to avoid any values of x that make the denominator equal to zero, as these values are undefined.
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