How to Find the Derivative of Cot(x) using the Quotient Rule

d/dx(cot(x))

To find the derivative of cot(x) with respect to x, you can use the quotient rule

To find the derivative of cot(x) with respect to x, you can use the quotient rule.

The quotient rule states that if you have a function of the form f(x) = g(x)/h(x), where both g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:

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

Applying the quotient rule to cot(x), we have:

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

g(x) = cos(x)
h(x) = sin(x)

Now let’s find the derivatives of g(x) and h(x):

g'(x) = -sin(x) (derivative of cos(x) is -sin(x))
h'(x) = cos(x) (derivative of sin(x) is cos(x))

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

f'(x) = [(cos(x)*cos(x)) – (-sin(x)*sin(x))] / [sin(x)]^2
= (cos^2(x) + sin^2(x))/sin^2(x)

Recall the trigonometric identity that cos^2(x) + sin^2(x) = 1, so the expression simplifies to:

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

Using the reciprocal identity 1/sin(x) = csc(x), we can rewrite the derivative as:

f'(x) = csc^2(x)

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

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