Question Video: The Derivative of an Inverse Cotangent Function Mathematics

Video Transcript

Find d by d𝑥 of the inverse cot of 𝑥.

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

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

Returning to the point where cot 𝑦 was equal to 𝑥, we’ll now square both sides of this equation. This gives us cot squared 𝑦 is equal to 𝑥 squared. One of our trigonometrical identity states that cot squared 𝜃 plus one is equal to cosec squared 𝜃. Rearranging this, we see that cot squared 𝜃 is equal to cosec squared 𝜃 minus one.

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

d𝑦 by d𝑥 is equal to negative one over one plus 𝑥 squared. This is the derivative of the inverse cotangent function.