Derivative of cot(x) with respect to x – Explanation and Calculation using Quotient Rule with Simplifications

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 for two functions u(x) and v(x), the derivative of their quotient is given by:

(d/dx)(u(x) / v(x)) = (v(x) * u'(x) – u(x) * v'(x)) / [v(x)]^2

In this case, u(x) is 1 and v(x) is tan(x).

Let’s calculate the derivative using the quotient rule:

u'(x) = 0 (since the derivative of the constant function 1 is 0)
v'(x) = sec^2(x) (since the derivative of tan(x) is sec^2(x))

Using the quotient rule, we have:

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

Simplifying this expression, we get:

(d/dx)(cot(x)) = -sec^2(x) / tan^2(x)

Now, let’s simplify the expression further using trigonometric identities. Recall that the reciprocal identity for tangent is:

tan(x) = 1/cot(x)

Substituting this into the expression, we have:

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

Simplifying further, we get:

(d/dx)(cot(x)) = -sec^2(x) * cot^2(x)

So, the derivative of cot(x) with respect to x is -sec^2(x) * cot^2(x).

I hope this explanation helps! Let me know if you have any further questions.

More Answers: