Derivative of cot(x) using the quotient rule: Step-by-step explanation

𝑑/𝑑𝑥[cot 𝑥]

To find the derivative of cot(x), we can use the quotient rule

To find the derivative of cot(x), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then its derivative is given by:

f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]²

In this case, g(x) = 1 and h(x) = tan(x).

Therefore, g'(x) = 0 (the derivative of a constant is zero) and h'(x) = sec²(x) (the derivative of tan(x) is sec²(x)).

Now, let’s plug these values into the quotient rule formula:

f'(x) = [0 * tan(x) – 1 * sec²(x)] / [tan²(x)]

Simplifying this expression further, we have:

f'(x) = -sec²(x) / tan²(x)

Since sec²(x) equals 1 + tan²(x), we can substitute this in the numerator:

f'(x) = -(1 + tan²(x)) / tan²(x)

Now, we can simplify this expression:

f'(x) = -1 / tan²(x) – 1

Finally, we can rewrite -1 / tan²(x) as -cot²(x):

f'(x) = -cot²(x) – 1

Therefore, the derivative of cot(x) is -cot²(x) – 1.

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