d/dx cot(x)
To find the derivative of the cotangent function, we can use the quotient rule of differentiation
To find the derivative of the cotangent function, we can use the quotient rule of differentiation. The cotangent function can also be written as the ratio of cosine and sine functions, i.e., cot(x) = cos(x) / sin(x).
Let’s differentiate cot(x). We’ll use the quotient rule, which states that if we have a function f(x) = g(x)/h(x), then the derivative f'(x) is given by:
f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
For cot(x) = cos(x) / sin(x), we can identify g(x) = cos(x) and h(x) = sin(x).
Taking the derivatives of g(x) and h(x), we have:
g'(x) = -sin(x)
h'(x) = cos(x)
Plugging these values into the quotient rule, we get:
cot'(x) = [(-sin(x)) * sin(x) – cos(x) * cos(x)] / [sin(x)]^2
Expanding and simplifying this expression further, we have:
cot'(x) = [-sin^2(x) – cos^2(x)] / [sin^2(x)]
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can simplify the numerator:
cot'(x) = -1 / [sin^2(x)]
Hence, the derivative of cot(x) is -1 / [sin^2(x)].
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