Applying the Quotient Rule: Finding the Derivative of cot(x) with Respect to x

d/dx(cotx)

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

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

The quotient rule states that if we have a function f(x) = g(x) / h(x), then the derivative of f(x) is given by:

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

Let’s apply this rule to cot(x).

In this case, g(x) = 1 and h(x) = tan(x).

First, we need to find the derivative of g(x):

g'(x) = 0 (the derivative of a constant is zero).

Next, we need to find the derivative of h(x):

h'(x) = sec^2(x) (the derivative of tan(x) is sec^2(x)).

Now, we can use the quotient rule:

cot'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2
= [0 * tan(x) – 1 * sec^2(x)] / [tan(x)]^2
= -sec^2(x) / [tan(x)]^2

We can simplify further using trigonometric identities. Remember that sec^2(x) = 1 + tan^2(x).

cot'(x) = -sec^2(x) / [tan(x)]^2
= -(1 + tan^2(x)) / [tan(x)]^2
= -1/[sin^2(x)/cos^2(x)]
= -cos^2(x)/sin^2(x)
= -cos^2(x)/[1-cos^2(x)]
= -cos^2(x)/(1-cos^2(x))
= -1

Therefore, the derivative of cot(x) with respect to x is -1.

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