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. The derivative of cot(x) can be written as:
f'(x) = (g(x) * f'(x) – f(x) * g'(x)) / (g(x))^2
where f(x) is the numerator and g(x) is the denominator of the cot(x) function. By representing cot(x) as the quotient of cos(x) divided by sin(x), we can find the derivatives of cos(x) and sin(x) and substitute them in the quotient rule formula.
cos(x) and sin(x) are both functions whose derivatives we know. The derivative of cos(x) is -sin(x) and the derivative of sin(x) is cos(x). Hence, f'(x) = -sin(x) and g'(x) = cos(x).
Substituting these values into the quotient rule formula:
f'(x) = (g(x) * f'(x) – f(x) * g'(x)) / (g(x))^2
= (sin(x) * (-sin(x)) – cos(x) * cos(x)) / (sin(x))^2
= (-sin^2(x) – cos^2(x)) / sin^2(x)
= -1 / sin^2(x)
Simplifying the expression, we have:
f'(x) = -cosec^2(x)
Therefore, the derivative of cot(x) with respect to x is -cosec^2(x).
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