Derivative of Cot(x): Calculating the Derivative of Cotangent with Respect to x

d/dx(cotx)

To find the derivative of cot(x) with respect to x, we can use the quotient rule

To find the derivative of cot(x) with respect to x, we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = u(x)/v(x), the derivative of f(x) is given by:

f'(x) = (v(x)u'(x) – u(x)v'(x)) / (v(x))^2

In this case, u(x) = 1 and v(x) = tan(x). Therefore, applying the quotient rule, we have:

cot(x) = cos(x) / sin(x)

Now, let’s calculate the derivatives of u(x) and v(x) separately:

u(x) = 1, so u'(x) = 0 (the derivative of a constant is zero)

v(x) = tan(x), so v'(x) = sec^2(x) (using the derivative of tan(x) formula)

Now, substitute the values into the quotient rule formula:

f'(x) = ((tan(x))(0) – (1)(sec^2(x))) / (tan(x))^2
= -sec^2(x) / tan^2(x)

Since sec^2(x) is equal to 1 + tan^2(x), we can rewrite the expression as:

f'(x) = -1 / (1 + tan^2(x))

Recall that 1 + tan^2(x) is equal to sec^2(x), so we can simplify further:

f'(x) = -1 / sec^2(x)

Since sec^2(x) is equal to 1 / cos^2(x), we can rewrite the expression as:

f'(x) = -cos^2(x)

Therefore, the derivative of cot(x) with respect to x is -cos^2(x).

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