The Quotient Rule: Derivative Of Cot(X) Explained Step By Step

derv cot(x)

-(csc^2(x))

The derivative of cot(x) can be found using the quotient rule of differentiation, since cot(x) can be written as cos(x)/sin(x). The quotient rule states:

If y = u/v, where u(y) and v(y) are both functions of y, then

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

Applying this to cot(x), we get:

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

Let u = cos(x) and v = sin(x). Then,

du/dx = -sin(x) (derivative of cos(x))
dv/dx = cos(x) (derivative of sin(x))

Plugging these values into the quotient rule formula, we get:

d(cot(x))/dx = [(sin(x)*(-sin(x))) – (cos(x)*cos(x))] / sin^2(x)

Simplifying,

d(cot(x))/dx = -[sin^2(x) + cos^2(x)] / sin^2(x)

Recall that sin^2(x) + cos^2(x) = 1. Therefore,

d(cot(x))/dx = -1 / sin^2(x)

So, the derivative of cot(x) is -csc^2(x).

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