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Numerade Educator

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Problem 63 Easy Difficulty

Prove the formula for $ (d/dx) (\cos^{-1} x) $ by the same method as for $ (d/dx) (\sin^{-1} x). $

Answer

$$-\frac{1}{\sqrt{1-x^{2}}}$$

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Video Transcript

and this problem, we have to use our knowledge of implicit differentiation and the inverse of, um, pardon me, the derivatives of inverse trig functions To prove the derivative of a of a specific function that were given, Yeah, this will make more sense once we go through the process. So we're told that why equals the inverse co sign of X? So what does that mean? That means the coastline of y equals X. That's just a property of the inverse trig functions. So we need to prove the derivative, and we have to use implicit differentiation. So we'll take the derivative of each side of this equation. So that means that the derivative of the coastline of why would be equal to the derivative of X. So what we do, we would get negative sign. Why Times D y d X equals one again. This part was chain rule, so we'd have this d y dx here. So then we can get d y dx by itself because that's what we want. We want the derivative. So do what do I d? X would be equal to negative one over the sine of why now we could do a substitution. The sign of why is something that we know we can use the Pythagorean identity, the main trick identity that we know. So we can say that Dwight D X equals negative one over the square root of one minus the cosine squared of Why, yeah, now this work is a little bit tricky. We defined what? X waas. We said that the coastline of y equals X so we can do another substitution. Do I. D. X would be equal to negative one over the square root of one minus x squared. So then we can say that the derivative of the inverse co sign of X with respect to X is negative one over the square root of one minus X squared. And that was our goal. That's what the problem told us we needed to dio. So I hope that this problem helped to understand a little bit more about implicit differentiation. And also, I hope, is built your intuition of the differentiation rules of inverse trig functions