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Find the derivative of the following functions.$$f(x)= rac{x}{x+1}$$...

Question

Find the derivative of the following functions.$$f(x)= rac{x}{x+1}$$

Find the derivative of the following functions. $$f(x)=\frac{x}{x+1}$$



Answers

Derivatives Find and simplify the derivative of the following functions. $$f(x)=\frac{x}{x+1}$$

Let's find the derivative of this function F of X. That is, we're going to be finding f prime of X so we can apply the quotient rule to this. Therefore, we first have the bottom term. That is, each the X plus one multiplied by the derivative of the top term just into the X minus the top term either the ex times derivative of the bottom term, which is just you. The X derivative of 10 and derivative of the X is you the X and all this is divided by the bottom term squared. That is, either the X plus one all squared. Remember those parentheses. They're very important. Now we have to simplify this. So first off, we can distribute through these parentheses, giving us E to the X squared, or either the two x plus ive the X minus another each the X squared or e to the two X, and this is divided by well, squaring this out, we would get e to the two X or you the X squared plus to eat the X plus one. Now, there's some more simplification to be done here. That is, as you can see, we can cancel on E to the two X and a negative each of the two x from the top, leaving us with just need to the X on the top and then on the bottom. We can't simplify any further, so we just have each the two x plus two e to the X plus one. Thus, since we can't simplify any further, this is our final answer.

Okay, so we're finding the derivative of this function now as an exercise. Since this is clearly the product of two functions, you should go ahead and use the product room. But I'm actually just going to expand this and take the derivative of it. The reason being, I know that this one over X squared term is gonna cancel out with this extreme term. And I know that for the most part, this is gonna look pretty nice. So I'm actually gonna rewrite it. Uh, with these two functions flipped since we're multiplying them and multiplication Kenyan community, if we can do that. And this is, ah, distributed law, Right? So this term I distributed is going to be X squared, plus one. Why? Because X squared times one is X squared and X squared, divided by X squared is one. And then I'm gonna have one plus one over X squared. This is still my function equals. I haven't done any calculus yet. We can simplify this as X squared, plus two, and then I'm gonna rewrite this as X to the negative to power, right? I can, of course. You know, I can flip the numerator and denominator. I just have to make sure to negate the exponents. Okay, so now I'm gonna take the derivative of this, Which is much easier, in my opinion, because this is a power rule. I get two X. This is a constant. It's derivative zero. And this is another powerful. This is, of course, minus two X to the minus two minus one. I always subtract one for my new X for my exponents. I'm left with two X minus two x to the minus three, which you can simplify if you want as two X minus two over. Execute.

So the question assets to find the derivative of seeking of X plus Cosi can't of X And before we begin, we can rewrite this in terms of signing co sign So we'll have one over co sign of X plus one over sign of X. And now because we have fractions, we can use the quotient rule. So we'll have why prime equals the derivative of the numerator. So zero times the denominator so co signer X minus the numerator times the derivative of the denominator minus sign of eggs all over the denominator squared. And this simplifies our first term will cancel out. We can distribute this negative, so that becomes Plus, So this first term simplifies to sign of X over co sine squared of X and we're just gonna highlight that below. Remember, we still have to do the quotient rule for this second term Here, one over sign of X So summer similarly will take the derivatives of the new married er zero times the denominator sign of X minus the numerator. So one times the denominator who sign of X all divided by sine squared Thanks and likewise this first term will cancel and we're left with co sign of X over science weird of acts. And now we can bring these two terms back together. So why prime equals sign of X over co sine squared of X minus co sign of X over sine squared and X And now we're gonna play around the little with our trigonometry. So for this first term, we get sign of X over co sign of X times one over Coastline of Axe. What we did is we broke this co sign apart minus and we're gonna do the same thing with this second term. So co sign of X over sign of X times one over sign of X. So we broke this denominator Signed Square Divac's And this can also be river in as tangent of X And this term can be written as c can't of X and now this term can lever in as co tangent of X. And this final term can be written as kusi can't have X and that yes, Allah derivative

Find the derivative of codeine Gen X over one plus Cosi can't attacks. And because this is a fraction, we have to use the Koshi rule and what I did was already defined our G on F terms. So your G term is always going to be what's aimed. The denominator on your F term will always be with us in the numerator, so you can begin now by solving for our derivatives will pull those terms into the question rule formula. So we have one plus kusi can't of X sometimes negative Cosi can't square to backs minus negative. Kroc can't Exco Tangent X Times Co tangent of X all over one plus e cosi can't squared of X Way like that and what we can do is now distribute that cozy can't squared of X term. And we have two negatives here, which will give us a positive so we'll get negative coast. He can't square two x minus. Cosi can't cute of X close Cosi can't x co tangent squared of X all over one plus lips one plus Cosi can't x squared. And when you're working with economic functions, it's always easier to you. Everything in terms of Simon Co sign cause it's more likely for things to cancel out. So what we'll do is rewrite everything in terms of that's will have negative one over sine squared of eggs minus one over sign cubed of X plus one over sign of x Times co sine squared of X over sine squared of X oh, over one plus one over sine squared of X And what you'll see now is that right here we have a sign times a sine squared so we can rewrite this denominator as sign Keep Rex and we have a sign cubed Alexe here so we can on these two terms together. So we'll get negative one over sign squared of X minus one plus co sign squared of eggs over saying Q two X over one plus one over sine squared and X And what will notice now is this right Here is a trigger gnomic identity. So we know that sine squared of X plus co sine squared of X equals one and therefore science squared of X equals one minus Come science where to vex So we can rewrite this as minus one over signed squared of X plus I sine squared of X over Sign key, Divac's over one plus one over sign squared backs. And now we can simplify this term to give us one over. It's not in squared of X plus one over sign of eggs over one plus one over sign square necks. And because we have a sign amble terms, we can act out a one over sine squared so we'll do one over sign of X. And what we're left with is one is minus one over a sign of X plus one over one plus one over sine squared X, and we can move, not negative out. And what will notice is that these two terms are the same. So we can cancel out one of these with that term and now will be left with one minus one over Sina packs over one plus one over Sign of X. And now we can rewrite thes as minus Cosi can't X over one plus close seeking of acts, which is our final answer


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