cot(x) derivative
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 u(x) = f(x)/g(x), then the derivative of u(x) is given by:
u'(x) = (f'(x) * g(x) – f(x) * g'(x))/(g(x))^2
In the case of cot(x), we can rewrite it as cot(x) = cos(x)/sin(x). Now, applying the quotient rule to this expression, we have:
cot'(x) = ((cos'(x) * sin(x) – cos(x) * sin'(x))/(sin(x))^2
The derivatives of cos(x) and sin(x) are well-known:
cos'(x) = -sin(x)
sin'(x) = cos(x)
Substituting these derivatives into the quotient rule formula, we get:
cot'(x) = ((-sin(x) * sin(x) – cos(x) * cos(x))/(sin(x))^2
Simplifying further, we have:
cot'(x) = (-(sin^2(x) + cos^2(x)))/(sin^2(x))
Now, recall the trigonometric identity sin^2(x) + cos^2(x) = 1. Using this identity, we can rewrite the derivative as:
cot'(x) = -1/(sin^2(x))
Therefore, the derivative of cot(x) is -1/(sin^2(x)).
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