Derivatives

In Exercises 27–...

Question

Derivatives

In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the derivative of the function F X equals X X divided by X 4 minus 1. A says it's x 5 plus X minus X X minus 3 x 4th sin X minus sine X all divided by X 4th minus 1. B says it's X 5th X minus X X plus 3 X 4th sin x plus sine x all divided by X 4 minus 1 squared. C says it's X 5 X minus X X minus 3 x 4th sin x minus sine X, all divided by X 4 minus 1 squared and the D says it's X 5 X minus X X minus 3 x 4th sin x minus X divided by X 4 minus 1 squared. Now how are we going to differentiate FFX? Well, notice here that FFX is a quotient of two functions. So to differentiate a quotient, we can apply the quotient rule. Recall that the quotient rule basically tells us that if we have a function Y that's equal to U divided by V where U is the numerator and V is the denominator. And they are both functions of X. Then the derivative of Y with respect to X equals U V minus VU. Let me just write that in a different way. Minusuv divided by v squared where U and V are the derivatives of U and V respectively. So for our problem If we let you be the numerator X sine x. And V be our denominator X the 4th minus 1. Once we differentiate these with respect to X, then we should be able to figure out the derivative of F of X. Now, let's first start by differentiating U. It's X X X. It's a product. So to differentiate it, we can use the product rule. And that product rule basically tells us that we differentiate both terms with respect to X and multiply them by the opposite, that is the term by the opposite derivative and then find their sum. So no. OK, when we differentiate here, we should find that we get you to be X plus X plus sine x, OK? X X plus X On the other hand, the derivative of V is going to be equal to 4 excubed. Now that we have the derivatives, we can apply the quotient rule. Because no, this means the derivative of FFX then is going to be equal to VU. OK. So that's going to be X 4 minus 1 multiplied by X plus X plus sine X. OK. Minus UV. OK, so that's gonna be X sine x multiplied by 4 X cube. I know all of this is gonna be divided by X 4 minus 1 V squared. No, we can simplify. First, let's start by expanding our terms. Let's expand the term in the first, let's expand the first term in our numerator, the first expression. Now, when we expand here, we're gonna get X to the 5th plus X, OK? minus X + X. Plus x to the 4th sin x minus sine X, OK? And now when we expand X X X multiplied by 4 X cubed, it's going to be 4 X to the 4th sin x. So we have all of this in our numerator and it's being divided by X 4 minus 12. No, can we simplify this any further? Well here if we think about it, OK. If we Take a good look, OK? Notice here that we have two terms that are X to the 4th sin. We have X to the 4th sin and minus 4 X to the 4th sin X. So if we subtract 4 to 4 X to the 4th sin from X to the 4th sin x, then we're going to end up with minus 3 X to the 4th sin. So then our quotient will simplified to X to the 5th plus X minus X + X minus 3 X to the 4th sin X. Minus sin x. That's all divided by, OK, this is all divided by X 4 minus 1 squared. We cannot simplify it any further, so this is the derivative of F of X. If we look back on our answer choices, that tells us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Derivatives

In Exercises 27–32, find dp/dq.

p = (q sin q) / (q² − 1)

Key Concepts

Derivatives

Quotient Rule

Trigonometric Functions

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