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~/1 POINTSTANAPCALC1O 5.5.030.Find the derivative of the function_9(t)9 (()Need Help?RaadtMatch IuJeltealuerInfezt...

Question

~/1 POINTSTANAPCALC1O 5.5.030.Find the derivative of the function_9(t)9 (()Need Help?RaadtMatch IuJeltealuerInfezt

~/1 POINTS TANAPCALC1O 5.5.030. Find the derivative of the function_ 9(t) 9 (() Need Help? Raadt Match Iu Jeltealuer Infezt



Answers

Use the quotient rule to find the derivative of the following.
$$y=\frac{9-7 t}{1-t}$$

All right. Hey, this problem we wish to find the derivative F fry backs for f of X equals six. Route X 24 X cubed plus that This question tells the understanding of differentiation. A particular is telling your knowledge of the derivative shortcuts we learn in this section. These are the five rules of differentiation list here constantly. All the power rule, the constant multiple rule that some difference rule in the exponential derivative. So in order to solve we need to rewrite F vaccine is most easily defensible form here that six extra one half. Because now we see how we use the power over the first term minus four X. Cubed plus nine. For each of the individual terms. We talked about this on different rules differentiate separately. Then we apply either Rules two and three or rule one to differentiate. Thus applying rule four more three. We have a prime Mexico 66 to 1 half minus four D. X cubed plus TX nine. Applying rules one and two gives six times one half X negative one half minus four times to execute or squared plus zero. This simplifies that F. Prime Mexico's three over X minus 12 X. Where

In this question. We're trying to take the derivative of the function y equals two x minus seven, all cubed. Since this is a composition, we're gonna use chain roll. So the first thing that we're gonna dio is we're gonna write why, as a composition f of G of X and F is gonna be our outside function, which in this case is gonna be X cubed, and we take the derivative we're gonna need the derivative of the function F itself, which is just a simple power rule his derivatives three x squared and our inside function G. We are going to choose to be just what's inside the parentheses for y two x minus said. And then we can take the derivative of G as well so that we can plug into the formula. In this case, we just get the constant too. So now if we want to find the derivative of why, since you've written as this composition, we know by the general that it's f prime of g of x times g prime. So when reforming this first term, we're gonna take f prime, compose it with G of X, so we know F prime is three x squared, but we're composing with G. So everywhere we see an X in the expression for F prime, we replace it with two X minus seven. This is gonna be three times two x minus seven all squared and they're gonna multiply by G prime, which is just too. We can simplify by multiplying our Constance. We get six times two X minus seven all squared.

We're gonna take the derivative of the polynomial X cubed minus 11 X squared plus seven x plus nine to take the derivative of these terms that are being added or subtracted will take the derivative of each term individually. It's this derivative rule to take the derivative of a power function, which is X to a constant exponents will multiply by the exponents and decrease the exponents by one. That's this rule and then to take the derivative of a constant number. The derivative of constant is zero. So then that's that rule taking the derivative will use the notation b Y t X taking a derivative of why, with respect to X, he's in the power rule in the first term multiplied by the exponents. Decrease the exponents by one gives us the derivative of X Cube registry explained, minus the derivative of 11 X squared, multiplied by the exponents two times 11 gives us 20 to increase The exponents by one gives us ex for the first and then plus the derivative of seven X is seven. We can think of that as an exponents of one. Well, then multiplying. One time seven gives us seven and then X to the zero, which is just gonna be one and then plus the derivative of a constant nine. And the derivative of Constant is here. So then we can write this derivative D Y de eggs as three X squared, minus 22 eggs plus seven x to the zero is just equal for one.

We're gonna take the derivative of this polynomial X cubed minus 11 X squared plus seven x plus nine to take the derivative oh, terms being added or subtracted would take the derivative term by term. That's this first rule to take the derivative of a power function which is X to a constant exponents would multiply by the exponents and increase the exploited by one sister. And then, if we want to take the derivative of the constant function, the derivative of the constant is zero. That's this rule that applying these rules, we use the D. I. D. X notation to show that this is the derivative. The derivative of X cube, multiplied by the exponents, decreased the exponents by one. So that gives us the derivative of X cube is three x squared, the derivative of 11 x squared, multiplied by the exponents two times 11 gives us 22 decrease the exponents by one. Is this then X to the first for seven x? We can think of that a seven extra the first. So when we multiplied by the exponents gives us one time seven, which is seven. Increasing the exponent by one gives us extra zero extra zero is just one. And then for the derivative of a constant nine, the derivative of the constant is zero. So then we can right this derivative as three Ex wed minus 22 x plus seven.


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