derivative of cot x
To find the derivative of cot(x), we will use the quotient rule
To find the derivative of cot(x), we will use the quotient rule. Recall that the quotient rule states that if you have a function f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x), denoted as f'(x), is given by (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2.
For cot(x), we can write it as the quotient of cosine and sine functions, cot(x) = cos(x)/sin(x).
Now, let’s find the derivatives of the numerator (cos(x)) and denominator (sin(x)).
Using the chain rule, the derivative of cos(x) is given by -sin(x).
Similarly, the derivative of sin(x) is cos(x).
Now, applying the quotient rule, we have:
cot'(x) = (cos'(x) * sin(x) – cos(x) * sin'(x)) / (sin(x))^2.
Using the derivatives we found earlier, cot'(x) = (-sin(x) * sin(x) – cos(x) * cos(x)) / (sin(x))^2.
Simplifying further, cot'(x) = (-sin^2(x) – cos^2(x)) / (sin^2(x)).
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the derivative as cot'(x) = (-1) / (sin^2(x)).
Finally, we can also express the derivative in terms of cosec^2(x), since cot(x) is equivalent to 1 / tan(x), which is in turn equivalent to (1 / sin(x)) / (1 / cos(x)), and reciprocal of sin is cosec, so cot(x) = (1 / cos(x)) / (1 / sin(x)) = cosec(x) / cos(x).
Therefore, cot'(x) = (cosec^2(x)) / cos(x).
In summary, the derivative of cot(x) is -cosec^2(x) / sin^2(x), or equivalently, cosec^2(x) / cos(x).
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