7–14. Find the derivative the following ways:
a. Using t...

Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

f(w) = w³ -w / w

Accepted Answer

Hello, in this video, we are going to be finding the derivative of the given function by using the quotient rule. The function given to us is F of X is equal to 2 X2 minus 3X divided by X. Since we are given a rational function, we will go ahead and find the derivative by using the quotient rule. The quotient rule is defined as the following. If we want to take the derivative of a rational function, we will define the numerator and denominator as the respective functions F of X and G of X. Then the quotient rule is defined as the following. It will be G of X multiplied by the derivative F X minus F of X multiplied by the derivative G X divided by G of X squared. So, in order to use the quotient rule, let's go ahead and define our numerator and denominator as F vex and GovXx. In this case F of X for the quotient rule will be defined as 2 X2 minus 3 X. And the denominator will be defined as G of X, which is equal to X. Next, what we need to do is we need to take the derivatives of our numerator and denominator. The derivative of the numerator F of X will be defined as 4 X minus 3 by the power rule. We will also apply the power rule to the denominator, and the derivative of the denominator G of X will equal to 1. Now that we have the 4 pieces of information we need for the quotient rule, let's go ahead and find the derivative. By using the quotient rule, the derivative of our original function will be defined as G of X, which is X, multiplied by the derivative F of X, which is 4 X minus 3, minus the original F of X 2 X2 minus 3X, multiplied by the derivative G of X, which is 1, all divided by G of X2, which is X2. Now what we need to do is we need to go ahead and simplify the derivative. We can start by distributing X to the quantity 4 X minus 3. Now, 2 X2 minus 3X multiplied by 1, will just give us 2 X2 minus 3X, so we don't have to worry about the 1. But we do need to distribute this negative sign across 2X2 minus 3X. This will allow us to simplify the derivative to be 4 X2 minus 3X minus 2X2 plus 3X. And this will be divided by X squared. Now what we will do is we will go ahead and combine like terms. We can combine negative 3X and positive 3X, and that will cancel the terms out. 4 X2 minus 2 X2 will leave us with 2 X2. So the derivative will further simplify to be 2 X 2 divided by X2. Since we have a factor of X2 in both the numerator and denominator, we can simplify these quantities to be 1, and that will leave us with the value of 2, which is going to be the derivative of the original function F of X. So 2 is going to be the derivative of the given function. And with that being said, the solution to this problem is going to bed. So, I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w

Key Concepts

Product Rule

Quotient Rule

Simplification of Derivatives

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