(d/dx) cot(x)
To differentiate the function cot(x) with respect to x, we can use the quotient rule
To differentiate the function cot(x) with respect to x, we can use the quotient rule.
The quotient rule states that if we have a function u(x) divided by v(x), the derivative of this function is given by:
(d/dx) (u(x) / v(x)) = (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2
In this case, u(x) = 1 (since cot(x) can be written as 1/tan(x)) and v(x) = tan(x).
Next, we need to find the derivatives of u(x) and v(x):
u'(x) = 0 (since 1 is a constant and its derivative is zero)
v'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x))
Now we can plug these values into the quotient rule formula:
(d/dx) cot(x) = ((tan(x) * 0) – (1 * sec^2(x))) / (tan(x))^2
Simplifying this expression, we have:
(d/dx) cot(x) = -sec^2(x) / tan^2(x)
Now, using the trigonometric identity tan^2(x) + 1 = sec^2(x), we can further simplify:
(d/dx) cot(x) = -1 / (tan^2(x) * tan^2(x)) = -1 / tan^4(x)
Therefore, the derivative of cot(x) with respect to x is -1 / tan^4(x).
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