Understanding the Derivative of Cot(x) with Respect to x: Derivation Using the Chain Rule

d/dx(cot(x))

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

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

Recall that the cotangent function can be written as cot(x) = 1/tan(x).

Let’s start by finding the derivative of the tangent function, tan(x).

The derivative of tan(x) = sec^2(x), where sec(x) is the secant function.

Now, let’s use the chain rule to find the derivative of cot(x) = 1/tan(x).

(d/dx)(cot(x)) = (d/dx)(1/tan(x))

To find the derivative of 1/tan(x), we need to use the quotient rule, which states that if we have a function of the form f(x) = g(x)/h(x), then the derivative is given by:

(d/dx)(f(x)) = (h(x)*(d/dx)(g(x)) – g(x)*(d/dx)(h(x))) / (h(x))^2

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

Using the quotient rule, we can find the derivative as follows:

(d/dx)(1/tan(x)) = (tan(x)*(d/dx)(1) – 1*(d/dx)(tan(x))) / (tan(x))^2

Since (d/dx)(1) = 0, we can simplify this expression:

(d/dx)(1/tan(x)) = – (d/dx)(tan(x)) / (tan(x))^2

Now, substituting the derivative of tan(x) = sec^2(x) into the expression:

(d/dx)(1/tan(x)) = – sec^2(x) / (tan(x))^2

And since cot(x) = 1/tan(x), we can write it as:

(d/dx)(cot(x)) = – sec^2(x) / (cot(x))^2

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

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