Learn How To Find The Derivative Of Cotangent Of X Using Quotient Rule Of Differentiation

d/dx cot x

-csc^2 x

Let us use the quotient rule of differentiation to find the derivative of the cotangent of x with respect to x.

Recall that the quotient rule states that for u = f(x) and v = g(x), the derivative of u/v with respect to x is:

(d/dx) [u(x)/v(x)] = [v(x)*(du/dx) – u(x)*(dv/dx)] / [v(x)]^2

In this case, we can let u = 1 (since cot x = 1/tan x) and v = tan x. Therefore:

(d/dx) [cot x] = (d/dx) [1/tan x]
= [-1*(d/dx) [tan x]] / [tan x]^2

Next, we need to find the derivative of tangent of x with respect to x, which is:

(d/dx) [tan x] = sec^2x

Substituting this back into our earlier equation, we get:

(d/dx) [cot x] = [-1*sec^2x] / [tan x]^2

Now, using the trigonometric identity tan^2x + 1 = sec^2x, we can simplify this expression as follows:

(d/dx) [cot x] = [-1*(1 + tan^2x)] / [tan x]^2
= -[1/tan^2x + 1]

Therefore, the derivative of cotangent of x with respect to x is:

(d/dx) [cot x] = -cosec^2x.

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