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

Find d/dπ‘₯ cos⁻¹ π‘₯.

02:49

Video Transcript

Find d by dπ‘₯ of the inverse cos of π‘₯.

In this question, we need to differentiate the inverse cos or arccos of π‘₯ with respect to π‘₯. We begin by letting 𝑦 equal the inverse cos of π‘₯. Taking the cos or cosine of both sides of this equation gives us cos 𝑦 is equal to π‘₯. We will then differentiate both sides of this equation with respect to π‘₯. We know that differentiating cos π‘₯ with respect to π‘₯ gives us negative sin π‘₯. Using our knowledge of implicit differentiation, the left-hand side becomes negative sin 𝑦 multiplied by d𝑦 by dπ‘₯. Differentiating the right-hand side simply gives us one. We can then divide both sides of this equation by negative sin 𝑦 such that d𝑦 by dπ‘₯ is equal to negative one over sin 𝑦.

Whilst this is an expression for the derivative, it is not in terms of π‘₯. We will go back to the point where cos 𝑦 is equal to π‘₯ and square both sides. This gives us cos squared 𝑦 is equal to π‘₯ squared. One of our trigonometrical identities states that sin squared πœƒ plus cos squared πœƒ is equal to one. This means that cos squared πœƒ is equal to one minus sin squared πœƒ. cos squared 𝑦 is therefore equal to one minus sin squared 𝑦, which we know is equal to π‘₯ squared. This equation can be rearranged so that one minus π‘₯ squared is equal to sin squared 𝑦. If we square root both sides of this equation, we see that sin 𝑦 is equal to the square root of one minus π‘₯ squared.

We can now substitute this into the denominator of our expression for d𝑦 by dπ‘₯. d𝑦 by dπ‘₯ is equal to negative one over the square root of one minus π‘₯ squared. This is the derivative of inverse cos of π‘₯. There is one more thing we need to be careful of as the square root of any negative number is not real. This means that this is only true for values of π‘₯ greater than negative one and less than one. π‘₯ cannot be equal to these values as this would leave us with the square root of zero on the denominator, which is undefined.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.