### Video Transcript

Find d by dπ₯ of the inverse cot of π₯.

In this question, we need to find the derivative of the inverse of cot π₯ with respect to π₯. We begin by letting π¦ equal the inverse of cot π₯. Taking cot or the cotangent of both sides of this equation gives us cot π¦ is equal to π₯. Our next step is to differentiate both sides of this equation with respect to π₯. We know that differentiating cot π₯ with respect to π₯ gives us negative cosec squared π₯.

Using our knowledge of implicit differentiation, differentiating cot π¦ with respect to π₯ gives us negative cosec squared π¦ multiplied by dπ¦ by dπ₯. Differentiating π₯ on the right-hand side gives us one. We can then divide both sides of this equation by negative cosec squared π¦ so that dπ¦ by dπ₯ is equal to negative one over cosec squared π¦. Whilst we do have an expression for dπ¦ by dπ₯, this is not in terms of π₯.

Returning to the point where cot π¦ was equal to π₯, weβll now square both sides of this equation. This gives us cot squared π¦ is equal to π₯ squared. One of our trigonometrical identity states that cot squared π plus one is equal to cosec squared π. Rearranging this, we see that cot squared π is equal to cosec squared π minus one.

This means that cot squared π¦ is equal to cosec squared π¦ minus one. And we know this is equal to π₯ squared. We can then add one to both sides of this equation so that cosec squared π¦ is equal to one plus π₯ squared. Finally, we substitute this into the denominator in our expression for dπ¦ by dπ₯.

dπ¦ by dπ₯ is equal to negative one over one plus π₯ squared. This is the derivative of the inverse cotangent function.