The Quotient Rule: Differentiating cot(x) with Respect to x

d/dx cotx

To differentiate cot(x) with respect to x, we can use the quotient rule

To differentiate cot(x) with respect to x, we can use the quotient rule. The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient (u(x)/v(x)) is given by (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2.

For cot(x), we can rewrite it as 1/tan(x) since cot(x) is the reciprocal of tan(x). Therefore, we have:

cot(x) = 1/tan(x)

Using the quotient rule, we can find the derivative of cot(x) as follows:

u(x) = 1
v(x) = tan(x)

Finding the derivatives of u(x) and v(x):

u'(x) = 0 (the derivative of a constant 1 is zero)
v'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x))

Now applying the quotient rule:

cot'(x) = (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2

cot'(x) = (tan(x) * 0 – 1 * sec^2(x)) / (tan(x))^2

cot'(x) = -sec^2(x) / tan^2(x)

We can simplify further using the fact that sec^2(x) = 1 + tan^2(x). Substitute this identity:

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

Therefore, the derivative of cot(x) with respect to x is -1 / (1 + tan^2(x)).

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