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Differentiate the function using one or more of the differentiation rules. y=(4x+5)8...

Question

Differentiate the function using one or more of the differentiation rules. y=(4x+5)8

Differentiate the function using one or more of the differentiation rules. y=(4x+5)8



Answers

Differentiate the functions using one or more of the differentiation rules discussed thus far. $$y=\left(\frac{x}{x^{2}+1}\right)^{2}$$

Look. So this is the first time we're actually doing the chain rule in this section X squared plus five to the 15th Power. And when I'm teaching the chain rule, I tell students. So look for that inter function. So I mean, just underlining the inter function. And then the outer function is taken to the 15th power. So as I do in the derivative, I like this notation My i d y d x, or you might have y prime. What I'm All I'm doing is the derivative of the outer function First, which is bringing that 15 in front and then it goes to the 14th power. What? What you do then, is you leave the inter function alone and you multiply by the derivative of the inter function, while the derivative of X squared is two x and the director of a 50 So no need to write down plus zero. Now most people would that rewrite this problem because multiplication is associative community that you could move that two x in front times 15 is 30 x x squared plus by to the 14th power is the same. That is your answer

Were given this function and I'm we're going to do the product rule. Here's your product, but we also have the chain rule embedded within this product rule. So as you recalled the product rule, you take the derivative of the left side of that product. What should give me 15 x squared? Remember, bring that explanation front, multiply it, subtract one from your ex opponent. Leave the right side of your product alone to my sex to the fourth power and then plus now leave the left side alone and then take the derivative of the right side. Just change colors, which is your chain rule, where you bring the four in front. Tu minus X is now to the third power. Remember, you take that inter function stays the same and then times the derivative of the inter function, which is negative one now, I would simplify this on Lee a little bit. You know, you could factor things out, but I'm not going to do that. Uh, your teacher might ask you to. It's not very difficult. It's just personal preference. So positive test positive times. This negative changes a sign of that. So now it's 20 X Q. Five times four is 20 and that to minus access to the third power. And this is an acceptable answer, so I'm stopping here.

All right, So we're giving this function. Why goes to X cubed minus one times? That's a cube right there. Three X squared, plus one to the fourth. Now, when I'm doing this problem, I look for things. I see a giant product in here. That's my clue that we're using the product rule. So the derivative D Y x, uh, do the derivative of the left side. Well, that too is still there. The drift of of X cubed minus one is three x squared. The drift of of negative one is zero. Leave the right side alone so that three x squared plus one is still to the fourth power. Then plus this part of the product rule you leave the left side alone two and then execute minus one and then the right sides derivative is the chain rule where you bring that four in front. Now it's to the third power. Leave the function alone inside times the derivative of the inside, which would be six x again, the derivative of 10 And I wouldn't worry about trying to simplify this. I wouldn't worry about trying to factor things out. Um, I would just multiply some things together that are pretty obvious. Two times 36 Um, you know, here I'm looking at two times, four times six, you know, maybe do. Six times four is 24 times two is 48. 48. We have this X and just leave the other pieces alone. Don't even worry about distributing. You don't wanna Where you don't want to do something incorrect. You don't want to simplify incorrectly. So this is where I would stop. This is good. Yeah. Okay.

Eso. We have to find the derivative of this problem, and it's a giant giant product. So that's your clue that you're gonna do the product rule. Right? Here's your product. Two X minus one to the five force, then two X plus one to the three force power. So let's identify that product way on the left side. Eso when you do the derivative the product of sorry, the derivative of the left side Bring that five force in front well, to I mean, you could write this 10 force. Um, but that would reduce to five halves. And then that two x minus one would be you subtract one from the exponents. Um, so I'll be the 1/4 power 54 Spanish one is 1/4 times the derivative of the inside. So that's times to leave the right side alone. And I'm gonna do the next one in a different color. Plus, now the left side gets stays the same. 22 X minus one to the five force power. And now we have another chain where you bring that three force in front three. Force two x plus one is now. Subtract one from that exposure would be the negative 1/4 power times and derivative of the inside. I kind of ran out of room. There is times too, but this will simplify. Because if you notice we have a two times two, which would divide that for out and same thing here, I guess I just didn't show it that way. Now, I wouldn't worry about simplifying like, pull it like factoring anything. Um, I would just rewrite it by canceling out some of those terms, and I would not worry about a negative exponents. I know. Some other teachers would. I would not. Okay, Okay. And right here is your answer. Circled in green? Yeah.


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