Find the derivative the following ways:
Using the Produc...

Question

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

h(z) = (z³ + 4z² + z)(z - 1)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of FF T equals T cubed plus 2 T 2 minus T + 1 multiplied by 5T minus 4 using the product rule. For our answer choices, A says the derivative is 20 T cubed plus 18T2 minus 26 T + 9. B 16T cubed plus 15 T 2 minus 8 T + 6. C 20T cubed plus 18T2 minus 18 T + 6, and D16T 2 + 25T minus 3. Now how can we figure out the derivative using the product to? What does that rule say? Well, recall that what it basically tells us is that we, if we have a function F of T that's equal to UV where U and V are functions of T, then the derivative of F of T is going to be equal to the derivative of u multiplied by V plus the derivative of V multiplied by U. In other words, if we can differentiate both functions, we can multiply them by their original opposite function and then add. So let's go ahead and differentiate both our terms with respect to T. Let's let U equal TQ + 2 T 2 minus T + 1 and V be 5T minus 4. Now when you equals our first expression, OK, let me just write it out here. Then if we were supposed to differentiate, then the derivative of U with respect to T will be equal to 3 T squared. That's the derivative of T cubed plus 4 T, that's the derivative of 2 T squared minus 1, that's the derivative of negative T and plus 0. That's the the derivative of our constant 1, OK? So U or the derivative of U is 3T2 + 4T minus 1. How about V? V equals 5T minus 4. So that means that the derivative of V is going to be equal to 5. Now, since we have our derivatives of U and V and our original functions, then we can go ahead and differentiate FFT. Because no, this means then that the derivative of FFT is going to be equal to U. 3 T 2 + 40 minus 1. Multiplied by V 5 T minus 4. Plus V 5 multiplied by U T cubed plus 2 T squad minus T + 1. Now we can expand and simplify our expression. Let's go ahead and expand. No. If we multiply here, 3, 1st of all, let's multiply each term. Uh, so 3 T squared by this expression and then 40 and then 1, OK. So, that's 3 T squared multiplied by 5T 15 T cubed, uh, 40 multiplied by 5T, that's 20 T squared, and -1 multiplied by a 5T that's negative 5T. And no, 3 T squared multiplied by -4. OK, that's negative 12 T squared. -12 T squared, uh, 40 multiplied by -4, that's -16 T and -1 multiplied by -4, that's plus 4. If we expand our next term, 5 multiplied by T cubed, 5 T cubed, 5 multiplied by 2 T squared, 10 T squared, 5 multiplied by negative T-5T, and 5 multiplied by 15. Now we can simplify by grouping like terms, because 15 T cubed plus 5 T cubed equals 20 T cubed, OK. And next 20 T2 minus 12 T2 plus 10 T2 equals 18 T2. OK. Next for our T terms, -5T, minus 16T, minus 5T is -26T, and then 4 + 5 equals 9. Thus, the derivative of F of T is 20T cubed plus 18T2 minus 26 T plus 9. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find the derivative the following ways:
Using the Product Rule or the Quotient Rule. Simplify your result.
h(z) = (z³ + 4z² + z)(z - 1)

Key Concepts

Product Rule

Quotient Rule

Simplification of Derivatives

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