Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. d/dx(tan^−1 x) =sec² x

Accepted Answer

Hello. In this video, we are going to be determining whether the given statement is true or false. So we want to determine whether the derivative of cotangent inverse of X is equal to negative Cosequin squad of X. In order to approach this problem, let's first rewrite the original argument into a more identifiable form. So the original argument wants us to find the derivative of cotangent inverse of X, and we want to compare it to negative cosequent squared of X. By definition, the derivative of cotangent inverse X will be negative 1 divided by 1 + X2. And by using the Pythagorean identities, we can rewrite negative cosicin squared X as -1 divided by sin squared of X. Now let's go ahead and take a look at the independent domains. Since both of the functions are in rational functions, let's try to find restrictions of the domains. For the first function, we will take the denominator and set it equal to 0. In order to solve for X, we will subtract one to both sides, giving us X's equal to -1, and if we take the square roots of both sides, we will get X's equal to the square root of -1, which is the imaginary number I. What this tells us is that there is no restriction in the domain of the left hand function, and the domain of the derivative of cotangent inverse of X is going to be negative infinity to positive infinity. Now, what about the domain of the right hand function? Well, -1 divided by squared X, there are restrictions in the domain, because sine is zero whenever X is equal to some multiple of N multiplied by pi divided by 2. So, whenever we have a multiple of pi divided by 2, that is going to cause this denominator to be 0. That means that we're going to have vertical asymptotes. So that tells us there are differences in the domains, or the domains are different. Also, the range of cotangent inverse of X is from negative infinity to positive infinity. Furthermore, the range of Cossekin 2 X or negative cossein 2 X is going to be from 1 to positive infinity. So the two functions also have differences in their ranges. So, since there are differences in the domains and ranges, that means that the function or the original statement is not true, it is false. And with that being said, the solution to this problem is going to be D. So I hope this video helps you in approaching this type of problem, and I will go ahead and see you all in the next video.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. d/dx(tan^−1 x) =sec² x

Key Concepts

Derivative of Inverse Functions

Trigonometric Derivatives

Chain Rule

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