Find the first and second derivatives of the functions in Exer...

Question

Find the first and second derivatives of the functions in Exercises 33–38.

w = ((1 + 3z) / 3z) (3 − z)

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says determine the 1st and 2nd derivatives, and we're given the function F of X is equal to the quantity of 1 + 2 X in quantity, divided by X, and that fraction is multiplied by the quantity of 5 minus X in quantity. So we need to find the 1st and 2nd derivatives of our function F of X for this problem. To begin with, we're going to multiply out the two terms that are given in our function F of X. In doing so, we'll have F of X is equal to the quantity of 5 plus 9 X minus 2 X squared in quantity, divided by X. And so now we'll need to calculate the first derivative, so we're gonna be looking for F of X. And that's going to be equal to the derivative with respect to X. Of the quantity of 5 + 9 X minus 2 X squad in quantity, divided by X in quantity. And so, in order to take this derivative, we're gonna need to use our quotient rules, since we're taking a derivative of a fraction. So, recall for the quotient rule, that if we have a function Y that has the form, Y is equal to U of X, divided by V of X. Its derivative Y is equal to the quantity of V multiplied by U minus U multiplied by V in quantity, divided by V2d. So, in our case here, U is going to be equal to 5 + 9 X minus 2 X2, and V is going to be equal to X. And so for U, it's going to be equal to the derivative with respect to X. Of the quantity of 5 + 9 X minus 2 X squad in quantity. And to take this derivative, we'll need to use our constant multiple rule as well as our constant rule. And so taking this derivative, this is going to be equal to 9 minus 4 X. We also need to calculate V and that's going to be equal to the derivative with respect to X of X. And when we take this derivative, that's gonna be equal to one. So now we can apply our quotient rule to find our derivative F of X, and so that's going to be equal to the quantity of V, which is X, multiplied by u, which is the quantity of 9 minus 4 X in quantity. They'll have minus U, which is the quantity of 5 + 9 X minus 2 X squad in quantity, multiplied by V which is 1. And all of that is divided by V2d, which will be X2. So now we just need to simplify this expression. We'll begin by multiplying everything out in the numerator. So doing so, this is going to be equal to the quantity of 9 X minus 4 X squad, minus 5 minus 9 X plus 2 X 2 in quantity, and that is still all being divided by X2. And so now we just need to collect like terms, so this is going to be equal to. If you look at our learning terms, we'll have 9 X minus 9X which will give me zero. So looking at the quadratic terms, we'll have -4X2 plus 2X2 which will give me minus 2 X2. And then for the constant term we'll just have minus 5, and all that is still being divided by X squared. So now all we're going to do is divide each term by X2d. And so that means that our first derivative is going to be equal to -2, -5 divided by X2. So this is the answer to the first half of the problem. Now, for the second half, we need to calculate the 2nd derivative, so we need to look for a prime of X. And that's going to be equal to the derivative with respect to X of the quantity of -2, -5 multiplied by X-rays to the -2 power in quantity. And here we've just rewritten 1 divided by X2d as X-rays to the -2 power. The reason why we did this is so that we could use our constant multiple rule in order to take this derivative. So applying our constant rule and constant multiple rule for this, this is going to be equal to. 0 plus. 10 multiplied by X-rays to the minus 3 power. And then just rewriting this, this is going to be equal to 10 divided by X cubed. And so that is the answer for the second derivative. So just to recap, our first derivative F of X is equal to -2, -5 divided by X2. And our second derivative, F of X is equal to 10 divided by XQ. So those are going to be the answers for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Find the first and second derivatives of the functions in Exercises 33–38.

w = ((1 + 3z) / 3z) (3 − z)

Key Concepts

Product Rule

Quotient Rule

Chain Rule

Watch next