Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

Question

Accepted Answer

Welcome back, everyone. In this problem, we want to solve the derivative with respect to X of 2 X multiplied by the inverse of 6 X where X equals root 2. A says it's a 4th of pi + 2, B 1/2 of pi + 2, C half of pi, and D 14th of pi. Now, if you look at our problem here, basically, we want to find the derivative of 2 X multiplied by the inverse of 6 X and then substitute X equals root 2 to solve. Notice that our expression is our product. What do we know about differentiating products? We recall that we can use the product rule of differentiation, and that basically tells us if Y is equal to uv where U and V are differentiable, then the derivative of Y is going to be equal to uv plus UV where U and V are the derivatives of U and V respectively. No for our problem here, OK, if we let you. B equal to 2 X and V be equal to inverse of X, then we should be able to differentiate and apply the product rule. So let's go ahead and try to do that. Let's set U to be 2 X, OK. And V to be equal to 2 inverse of X. No, the derivative of 2 Xu is going to be 2 by the polar rule, and we also know that the derivative of sex inverse X is going to be equal to 1 divided by the absolute value of X multiplied by the square root of X2 minus 1. That's by definition for an absolute value of X greater than 1, OK. So now that we have our derivatives, then we can apply the product to it because by the product rule then that means the derivative. Uh, 2 X inverse X. Is going to be equal to UV. So that's 2 inverse X. OK. Plus UV, OK? So that's gonna be 2 X divided by the absolute value. Of X multiplied by the square root of X2 minus 1. Now remember we're trying to evaluate or we're trying to find a value. We're trying to solve here for when X equals root 2. So if we're gonna solve for that, the question we have to ask ourselves is which absolute value of X or which value of X are we going to use rather for our absolute value function. Well, X equals root 2, OK. And since X. equals root 2, where we know that root 2 is positive, it's greater than 0. OK, then we can take the absolute value of X to be positive. In other words, the absolute value of X to be X. With that in mind then This would mean that our derivative, OK, the derivative. I Let me write it here. OK, this means that our derivative then is gonna be equal to 2 inverse X. Plus 2 x divided by X multiplied by the square root of X2 minus 1 and then we can cancel X in our numerator and denominator for the second term to make it 2 inverse X, OK. Plus 2 divided by the square root of x2 minus 1. Now that we have the derivative, we can evaluate it at X equals the square root of 2. Because now, to evaluate it there we'd substitute that into our formula, OK, which means then that the derivative at X equals root 2 is going to be two sec inverse of root 2. Plus 2 divided by the square root of root 2 squad minus 1. Now, the principal value range for the inverse function of sick is between 0 and pi inclusive, excluding pi divided by 2. OK. So in this case, we know that the sick inverse of route 2. Is going to be equal to 1/4 of pi, OK. Now for a second term that will be 2 divided by the square root of root 2 squad, which is 2 minus 1. And if we simplify this some more, 2 multiplied by 4th of pi equals 1/2 of pi, 2 minus 1 equals 1, so that's 2 divided by a root 1, which is 1, and thus we get our answer to be 1/2 of pi plus 2. Therefore, when we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

Key Concepts

Derivative

Secant and Inverse Secant Functions

Product Rule

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