The Derivative of Cot(x): Finding the Differentiation of the Cotangent Function using the Quotient Rule

d/dx[cotx]

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

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

The quotient rule states that for any function u(x) divided by v(x), the derivative can be found as:

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

First, we will identify u(x) and v(x) for the function cot(x):

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

Next, we need to find du/dx and dv/dx:

du/dx = 0 (since u(x) = 1, the derivative of a constant is always zero)

To find dv/dx, we will use the chain rule. The derivative of tan(x) is sec^2(x), and since the derivative of x with respect to x is 1, we have:

dv/dx = sec^2(x)

Now, we can substitute the values into the quotient rule formula:

d(cot(x))/dx = (tan(x) * 0 – 1 * sec^2(x)) / (tan^2(x))
= -sec^2(x) / (tan^2(x))
= -1 / sin^2(x)

So, the derivative of cot(x) is -1 / sin^2(x).

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