Derivative of cot x
To find the derivative of the cotangent function, we can use the quotient rule
To find the derivative of the cotangent function, we can use the quotient rule. The quotient rule states that for two functions f(x) and g(x), the derivative of their quotient (f(x)/g(x)) is given by:
(f'(x)g(x) – g'(x)f(x))/(g(x))^2
In this case, let’s consider the function f(x) = 1 and g(x) = tan(x), since cot(x) can be expressed as 1/tan(x).
Using the quotient rule, we differentiate f(x) and g(x) separately to find the derivatives:
f'(x) = 0 (since f(x) = 1, its derivative is always zero regardless of x)
g'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x))
Now we can plug these values into the quotient rule formula:
(cot'(x))(tan(x))^2 = (0*tan(x) – sec^2(x)*1)/(tan(x))^2
Simplifying this expression, we get:
cot'(x) = -sec^2(x)/(tan(x))^2
Since sec^2(x) equals 1 + tan^2(x) by the Pythagorean identity, we can substitute this into the derivative:
cot'(x) = -(1 + tan^2(x))/(tan^2(x))
Finally, using the identity cot^2(x) = 1 + tan^2(x), we can rewrite the derivative in terms of cot(x):
cot'(x) = -cot^2(x)
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