d/dx cot(x)
-csc^2 (x)
To find the derivative of cot(x) with respect to x, we must use the quotient rule since cot(x) can be rewritten as cos(x)/sin(x).
Recall that the quotient rule states that the derivative of f(x)/g(x) is given by [g(x)f'(x) – f(x)g'(x)]/[g(x)]^2.
Therefore, applying the quotient rule to cot(x) = cos(x)/sin(x), we have:
d/dx cot(x) = [sin(x)(-sin(x)) – cos(x)(cos(x))]/[sin(x)]^2
Simplifying this expression, we get:
d/dx cot(x) = [-sin^2(x) – cos^2(x)]/[sin^2(x)]
Recall that sin^2(x) + cos^2(x) = 1. Therefore, we can substitute in 1 for sin^2(x) + cos^2(x) to obtain:
d/dx cot(x) = -1/[sin^2(x)]
Therefore, the derivative of cot(x) with respect to x is -csc^2(x).
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