Question Video: The Derivative of an Inverse Cotangent Function Mathematics • Higher Education

Find (d/dπ‘₯) cot⁻¹ π‘₯.

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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 𝑦 equal the inverse of cot π‘₯. Taking cot or the cotangent of both sides of this equation gives us cot 𝑦 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 𝑦 with respect to π‘₯ gives us negative cosec squared 𝑦 multiplied by d𝑦 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 𝑦 so that d𝑦 by dπ‘₯ is equal to negative one over cosec squared 𝑦. Whilst we do have an expression for d𝑦 by dπ‘₯, this is not in terms of π‘₯.

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

This means that cot squared 𝑦 is equal to cosec squared 𝑦 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 𝑦 is equal to one plus π‘₯ squared. Finally, we substitute this into the denominator in our expression for d𝑦 by dπ‘₯.

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

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