In Exercises 13–24, find the derivative of y with respect to t...

Question

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

23. y = (x²+1)sech(ln x)

(Hint: Before differentiating, express in terms of exponentials and simplify.)

Accepted Answer

Welcome back, everyone. Find the D Y divided by DX4 Y equals X to the power of 6 + 1 multiplied by sash of 3 LN X A 9 X 2 B 6 X 2 C 3 X2, and D 12 x 2. Let's begin this problem by simplifying the function. Let's remember the definition of sesh. If we have sesh of X, it is going to be equal to 2 divided by e to the power of X plus e to the power of negative X. So in this case, we have sesh. Of 3 LNX Which is going to be equal to 2 divided by e to the power of 3 LN X, plus e to the power of -3 LN X. And we can simplify using the properties of logarithms. Specifically, we're going to have two. In the numerator. In the denominator, we can write E to the power of LN of X cubed using the power rule of logarithms. Specifically, we can take our constant 3 and rewrite it as. An exponent of X, and we're going to apply the same property for the remaining term. So we're adding E to the power of LN of X to the power of -3. So now, let's go ahead and apply the next property of logarithms. E to the power of LN of X cubed is just X cubed itself, and eats the power of LN of X the power of -3 is X the power of -3. Now, what we're going to do from here is simply multiply both sides. By X cubed, so that we're going to get 2 X cubed and the numerator. In the denominator, we're going to get X to the power of 6 + 1, right? So, we now have An expression of sash of 3N of X. Notice that Y is equal to x the power of 6 + 1 multiplied by sesh. So we get X the power of 6 + 1 multiplied by 2 x cubed divided by x to the power of 6 + 1. And then we can clearly cancel out X the power of 6 + 1 to get a simpler expression of Y. It is basically 2 X cubed. So now we can find the derivative of Y easily. We can factor out the constant, which is 2, and multiply by the derivative of X cubed which can be found using the power rule. So we get 2 multiplied by 3 x 2, which is 6 x squad. And therefore, the final answer for this problem is option B. Thank you for watching.

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.23. y = (x²+1)sech(ln x)(Hint: Before differentiating, express in terms of exponentials and simplify.)

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In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
23. y = (x²+1)sech(ln x)
(Hint: Before differentiating, express in terms of exponentials and simplify.)