How to Find the Derivative of Cot(x) Using the Quotient Rule and Simplify

d/dx(cotx)

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

To find the derivative of the function f(x) = cot(x), we can use the quotient rule. The quotient rule states that for two functions u(x) and v(x), if f(x) = u(x)/v(x), then the derivative of f(x) can be calculated as:

f'(x) = (u'(x)v(x) – u(x)v'(x)) / (v(x))^2

For cot(x), we can rewrite it as the following:

cot(x) = cos(x) / sin(x)

Now, let’s differentiate using the quotient rule.

u(x) = cos(x)
v(x) = sin(x)

Differentiating u(x) with respect to x, we get:
u'(x) = -sin(x)

Differentiating v(x) with respect to x, we get:
v'(x) = cos(x)

Now, substituting the values into the quotient rule formula, we have:

f'(x) = (u'(x)v(x) – u(x)v'(x)) / (v(x))^2
= (-sin(x) * sin(x) – cos(x) * cos(x)) / (sin(x))^2
= (-sin^2(x) – cos^2(x)) / sin^2(x)

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify further:

f'(x) = (-1) / sin^2(x)
= -csc^2(x)

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

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