Use the Product Rule hnd Ihe dervative of ihe given function b. Fnd tne derivative by expanding tne procuct Mtst_f(x) = (* - 3K2x+4)Use the product mule lind tho de...

a. Use the Product Rule to find the derivative of the given function. Simplify your result. b. Find the derivative by expanding the product first. Verify that your answer agrees with part $(a)$ $$f(x)=(x-1)(3 x+4)$$

Okay, So for part A, whereas to use our product, do you get our product role? Let's take the derivative of this time first, so that's F prime of it. That's equal to one time three x was for and then plus the derivative of three experts. Four. So that's this three times X ray. It's one so multiplying that out we get three X plus for an N plus three. One is three so that you go to six x plus one and now report be you want to expand first, an intake are devoted. So expanding our function off of X that you put too three X squared plus four minus three x and n minus four. So this is equal to three x squared plus X minus four are not taking our through evidence. That's derivative of our first time. So two times three. That's six sex in plus one. Okay, so we see that are derivative is the same. If we use our product rule or when we expect

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Starting with part A. We will be using the product rule first. I have the product rule written off to the right hand side, so let's get started. I want to think of X minus one has effort backs and three X plus four as G effects just to follow along with the format of the product rule. Okay, Product rule says I need f prime of X first. So I'm taking the derivative of X minus one, which is just one. And then I need G of x three X plus four plus F of X X minus one and the derivative of G of acts, which is just three. Crais from here, any to simplify and I want to use the distributive property to distribute the three on the far right side, So I have three x plus four plus three X minus three. Next, I need to combine like terms six X plus one. Great. Let's move on to part B. Part B wants me to expand first, so let me rewrite down the problem. F of X is equal to X minus one times the quantity three X plus four. Here I am going to need to use the foil method great, and that tells me to multiply the first terms. Three x squared. Multiply the outside terms plus four eggs. Multiply the inside terms minus three x and multiply the Last terms, which is minus four. Now I need to combine my like terms three X squared plus X minus four And now I need to find the derivative So f prime of X, the derivative of three x squared multiply two times three that is six X and then the derivative of X is just one, and the dura tive of negative four is just zero. So again, my final answer here is six x plus one.

Hello. So here we have Y is equal to the quantity T squared plus 17 times the quantity three t minus four sort of differentiate. We can go ahead and use a product or the product of two things here. So we take the first which is T squared plus 17 times the derivative of the second. The derivative of the second is just going to be three times three. And then plus the second so plus three t minus four times the derivative of the first which is going to be a two T. Plus seven. So then we get here well through this times three gives us a three T squared and then um Plus 21 T. And then we get we distribute here we get a three T times 60. It's going to be a plus um six T squared. And then we have a 21 t minus 18. Um Which is going to be 21 -8. is what 21 2019 13? So we get a plus 13 T. And then my 97 times four is a minus 28. Okay. And we combine like terms here and we get our our derivative um or Y. Prime is going to be equal to nine T squared plus 30 40 And then -28. Okay. So there's our derivative um by the product role. If we want to expand it out first. Well we looked at um so why remember was equal to t squared plus 70 times three t minus four. So if we distributed first we would get that Y would be equal to when we get a three uh T cubed. And then we'd have minus 40 square plus 21 T squared, which would be a plus 17 T squared. And then we have a minus 28 T. And then we can differentiate now just turned by term and we get well, we get a nine t squared and then plus 17 times two is a plus 34 T. and the Derivative of Native 2080 is a -28. And we see we have the same derivative. If we go ahead and use the product rule, or if we just expand out first and then differentiate turn by turn, take care.