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.

g(s) = 4s³ - 8s² +4s / 4s

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the derivative of GFX is equal to 5 x 3 minus 10 x 2 plus 5X all divided by 5X using the quotient rule. Awesome. So it appears for this particular problem, we're asked to figure out what the derivative of GFX is using the quotient rule. So with that in mind, first off, we need to recall and use the quotient rule. So let us note and recall that the quotient rule states that D by DX of U divided by V is equal to V multiplied by U. Minus U multiplied by V. All divided by V squad. And for this particular problem, let's make a note here the state that you is going to be equal to 5 multiplied by X 3 or 5 X 3 minus 10 x 2 plus 5 X. and let us say that V is going to be equal to 5 X. Fantastic. So now, our next step that we need to take. we need to differentiate you, and to in order to do that, in order to differentiate you, we need to recall and use the power rule which states that D by DX. Of X to the power of n. is equal to N multiplied by X to the power of N minus 1. So differentiating U using the power rule, we will find that U, which indicates the the first derivative, so U equals D by DX 5 X 3 minus 10 x 2. Plus 5 X is equal to 5 multiplied by 3 X to the power of 3 minus 1 minus 10 multiplied by 2 X to the power of 2 minus 1. Plus 5 X to the power of 1 minus 1, which will mean that you is equal to 15 X to the power up to minus 20 X + 5. So now we need to differentiate V. So when we do that, we'll find that V is equal to D by DX of 5 X, which is equal to 5. So now we need to apply the quotient rule, so we could say that G of X is equal to 5 X multiplied by 15 X squad minus 20 X + 5. Minus 5 X to the power of 3 minus 10 X 2 plus 5 X multiplied by 5. All divided by 5 X to the power of 2. So, when we do a little bit of simplifying, we'll find that G of X is equal to 75 X 3 minus 100 x 2 plus 25. X minus 25, X 3 minus 50 X 2 plus 25. X All divided by 25 X to the power of 2. Fantastic. So now what we're trying to do is we're trying to simplify down this expression as far as it can go. So our next step is we need to take G of X is equal to 75 x 3 minus 100 x 2 plus 25 X minus 25 x 3. Plus 50 X to the 2 minus 25 X, all divided by 25 X 2. So, now we need to combine like terms and do some more simplifyings. We need to take G of X is equal to 50 x 3 minus 50 x 2, all divided by 25 X to the power of 2. And this is also equal to 50 X of the power of 2, when we factor out, multiplied by X minus 1 divided by 25 x 2. So this means when we. Factor out as far as we can and we simplify down to our most simple form, you will find that G of X, so G of X is equal to 2. Multiplied by X minus 1, which is also equal to G of X, which is equal to 2 X minus 2, and that is our final answer. We did it. So once again, our final answer is G of X is equal to 2 X minus 2, and that's when we distribute the 2 to our to our expression inside of the parentheses. So that means that our final answer will be 2 X minus 2. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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.
g(s) = 4s³ - 8s² +4s / 4s

Key Concepts

Product Rule

Quotient Rule

Simplification of Derivatives

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