Derivative of cot(x) with Respect to x | Step-by-Step Explanation and Simplified Formula

d/dx(cot(x))

To find the derivative of cot(x) with respect to x, let’s start by recalling the identity for cotangent:

cot(x) = 1/tan(x)

Now, we can rewrite the expression as:

cot(x) = (1/tan(x))

To find the derivative, we need to use the quotient rule

To find the derivative of cot(x) with respect to x, let’s start by recalling the identity for cotangent:

cot(x) = 1/tan(x)

Now, we can rewrite the expression as:

cot(x) = (1/tan(x))

To find the derivative, we need to use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative f'(x) is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In this case, g(x) = 1 and h(x) = tan(x). Differentiating these functions gives us:

g'(x) = 0 (since derivative of a constant is zero)
h'(x) = sec^2(x) (derivative of tan(x) is sec^2(x), which is the reciprocal of cos^2(x))

Now, using the quotient rule, we have:

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

Simplifying, we get:

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

But we know the relationship between sec(x) and tan(x):

sec^2(x) = 1 + tan^2(x)

So, substituting this in, we have:

cot'(x) = – (1 + tan^2(x)) / tan^2(x)

Further simplifying, we get:

cot'(x) = -1/tan^2(x) – 1

Therefore, the derivative of cot(x) with respect to x is:

cot'(x) = -1/tan^2(x) – 1

Note that the negative sign in the derivative indicates that the cotangent function is decreasing, or sloping downwards, as x increases.

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