š/šš„[cot š„]
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)]Ā²
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) = secĀ²(x) (the derivative of tan(x) is secĀ²(x)).
Now, let’s plug these values into the quotient rule formula:
f'(x) = [0 * tan(x) – 1 * secĀ²(x)] / [tanĀ²(x)]
Simplifying this expression further, we have:
f'(x) = -secĀ²(x) / tanĀ²(x)
Since secĀ²(x) equals 1 + tanĀ²(x), we can substitute this in the numerator:
f'(x) = -(1 + tanĀ²(x)) / tanĀ²(x)
Now, we can simplify this expression:
f'(x) = -1 / tanĀ²(x) – 1
Finally, we can rewrite -1 / tanĀ²(x) as -cotĀ²(x):
f'(x) = -cotĀ²(x) – 1
Therefore, the derivative of cot(x) is -cotĀ²(x) – 1.
More Answers:
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