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Use the Chain Rule t0 findwhere 2 =X siny,X=/" , and_ ly=515.Xcos(Type an expression using X and yas the variables:dy 254(Type an expression using (as Ihe vari...

Question

Use the Chain Rule t0 findwhere 2 =X siny,X=/" , and_ ly=515.Xcos(Type an expression using X and yas the variables:dy 254(Type an expression using (as Ihe variable )(Type an expression using tas Ihe variable )

Use the Chain Rule t0 find where 2 =X siny,X=/" , and_ ly=515. Xcos (Type an expression using X and yas the variables: dy 254 (Type an expression using (as Ihe variable ) (Type an expression using tas Ihe variable )



Answers

Use the Chain Rule to find the derivative of the following functions. $$y=\left(2 x^{6}-3 x^{3}+3\right)^{25}$$

In this question. We're going abandoned derivative on the exponential form. So each of the new derivative can equal to the new prime temps de to the new. And in this question, were given the why go to the the power of the two X square so we can identify this one will be the new. So by formula, the white prime echo thio thio X square prime times age two x square And now that you x square the river to get it, go to the four x times aged two X square.

And this problem we must define the derivative F. Prime of X. Using chain rule version to where we are given the function. Why is equal to the coast? You can't X squared plus one. This question is challenging understanding of differentiation, particularly how to utilize the chain rule. To differentiate version to the chain rule states that for Y equals F of G of X. Dy dx F prime of G of X times two X. So we combine the above with previous differentiation techniques to solve so why it was F. Of G. Of X. Has G of X equals X squared plus X. Thus F of G of X. Is Costa can of G of X. Now we utilize our previous differentiation technique should be in a prime and G pride. Thus we have by chicken after differentiation F prime of G of X is negative coast. You can explore one coach and an extra plus one by the power rule. And the some difference rule G prime X is simply to expose one. That's our solution is negative two X plus one. Cause you cant X squared plus one co tangent, X squared plus one. As in box at the bottom for dy dx

Eso to find these e d t We want to take B c d x times dx DT plus dizzy d y times b Y t t and so soloing for easy d x we get that that is gonna be one over two x squared. Plus why and sorry Should be for X, then e x p t is equal You one half even the native half no disease. Why is equal to 1/2 x squared plus y? And you I d t equal the two thirds you didn't A of one third and I'm down for moments that we can calculate all this up. So ZD sequel to Borax over two X squared plus y what has either the native one half plus one over two x squared. But why? Times two thirds do you do that? You want Terry? And so I'm gonna go ahead and do the 1/2 x squared plus y from my itself. I'm gonna go ahead and actually what that's gonna be, I would end up being to t close t two thirds some, uh, multiply this four acts, which would be the force word of X by the one half to the native half, and that's gonna actually just give me two Emma, the end of the two thirds. Do you do the night? One third. That is our final answer right there.

Okay, so first we're looking for TV X. Be equal T you times do you x plus 18 TV times d v d X wind up giving us one over e aren't you x plus you ve square tons DVDs, Which is before? Why no maker substitution, which is or or it's cute times to eggs plus needed X squared minus y squared over B squared, which is 16 x squared. Why did six turns for Why cubed? And we'll go ahead and simplify these so we'll get one over too cute. My guess X squared minus y squared of four x squared y to the third, which will result in two x squared minus X squared minus y squared over four it expired. Why cubed, which is equivalent to X squared and it's y squared all over four x squared like you'd and sorry, this should be a plus There. I didn't carry over my negative science that will be X squared plus y squared over four x squared y cube. And so now we'll go ahead and work on do you want? Which would be what e do you You know why Plus you you you times e you Why that's going to be equal to one over me times negative to wine plus made, If you will be squared times 12 x y Square then we'll go ahead and do our substitution. One of our four x y cube. Native to why plus Native X squared minus y squared over 16 x squared. Why didn't stick? Sometimes 12 x y squared. We'll head and simplify the you know any of us negative Want over two x y squared minus X squared three X squared. I asked three y squared over for X. Why did the fourth? Oh, and I won't go ahead most by this bike. Two y squared. So I have native Y squared minus X squared minus three y squared. Be a plus because we're distributing the native son and then we'll have over four x y to the fourth. I'll be equivalent two y squared minus X squared over four x y to the fourth. That's our final answer


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