Mastering the Derivative of Cot(x) | A Step-by-Step Guide

d/dx cotx

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

To find the derivative of cot(x), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) with respect to x (denoted as f'(x)) is given by (g'(x)h(x) – g(x)h'(x))/[h(x)]^2.

Using this rule, let’s find the derivative of cot(x):

Given: f(x) = cot(x)

Let g(x) = 1 and h(x) = tan(x).

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = 0 (since 1 is a constant)

h'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x), which is 1/cos^2(x))

Now, we can apply the quotient rule to find f'(x):

f'(x) = (g'(x)h(x) – g(x)h'(x))/[h(x)]^2
= (0 * tan(x) – 1 * sec^2(x))/[tan(x)]^2
= -sec^2(x)/tan^2(x)

But we know that sec^2(x) = 1 + tan^2(x). So we can substitute sec^2(x) in terms of tan^2(x):

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

Finally, we can simplify the expression further by using the Pythagorean identity for tangent: 1 + tan^2(x) = sec^2(x):

f'(x) = -[sec^2(x)/tan^2(x)]
= -[1/tan^2(x)]
= -cot^2(x)

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

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