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Find the derivative of the function.$$h(x)=ln left(2 x^{2}+1ight)$$...

Question

Find the derivative of the function.$$h(x)=ln left(2 x^{2}+1ight)$$

Find the derivative of the function. $$ h(x)=ln left(2 x^{2}+1 ight) $$



Answers

Find the derivative of the function. $$ h(x)=\ln \left(2 x^{2}+1\right) $$

Way are justifying their little eight, which is natural law of X plus, Very expert minus one. We're gonna do that using four months. Three age three points happen in your textbook, which reads that they're rich. Full function off a natural order sponsor. Ji is a food. They were off the function g divided by B function itself. So we need to find our function G in this case, which is X plus spur you and squared minus what? What else? It can be reckless. Ex waas expired plus one forward off one. Okay, uh, so let's find me. There are function keep, which is There were two of the first turn, which is one waas there was involved the second term Just one power experience. My form, uh, mindful supply by the veritable keener on sandwiches too. Oh, so the studio will cancel out. Right? And you have one waas X divided by a squared minus. What? So we're gonna play this guy right here, and therefore G affects working the use. This one Okay. Oh, so fine. Ex open equal to Poland, plus x y five spare boot expired. My phone divided eggs plus square root expired minus. What have this part is a little bit tricky. Now we're gonna do I have all the permits and dominated by ex spare rude experience minus one. So he have one plus exploited boxcar route expert minus one you wanted by? They're rude. Expert minus one Parentheses. Warn Waas X over square root, X squared minds. Right. So it is a pencil off with this one, and this will give us the answer, which is on over skirt ex. Very like this.

Okay. We're finding the derivative of y equals X times Ln of absolute value of two minus x squared, we don't need to worry about the absolute value too much. When we're finding the derivative of the algorithm. It just allows. Are derivatives have as big a domain as possible? Okay, so we're gonna need to use a chain role. Are the product rule here? The general, too. So we have above the product will fool first. So we have y prime equals ex prime times, Alan of two minus X squared plus x of Caroline of two minus X squared crime. Okay, well, the derivative ex prime is just one. So we just get Ln of two minus x squared here, plus x times. And now we need to use the chain Rule one over two, minus X squared times two minus x squared. Prime. Yeah. Okay, then finally we get Ellen of two minus x squared plus x over two. What is X squared times? Negative. Two x. So you can simplify that slightly and we write it as Ellen of two minus. I swear, plus or really should probably be minus two x squared over two minus x squared, which has a nice cemeteries to it.

Given the function X go to and land off Thanks. Plus square root of X squared minus one. And we need to find a hunch. Brand X here, recorded live in the airline on the function off the X and want to find a derivative we could, you know, you creme eggs. Over the years of X here we can identify This one will be the year off X Therefore, we can get in code you The express squared on the X Glamorous what? Minus one Bram Dividing by the X plus square root of X squared minus one. And now nobody, not the numerator. Then they drifted. Exhale we could you one plus that the rift Them This one we should get. We can now do it under side here so we can read Aziz experiments. £1 half I want to do that derivative, which you get a good job, huh? We'll bring it down. Times X squared minus one. Bounce off minus 1/2 by General win it your temps Duh. You drift other inside him, which is could you two x Therefore we can consider that you with that you and we have the X over and, uh, next square minus one less square It here never was getting now in cartoon. Uh, thanks off the square of the X squared minus one and known dividing by the X plus square The X squared minus one. And here we say we concerned if I on the numerator we can reach There s the square it on the X squared minus one plus X totally divided by the square than the X squared minus one. So we divide by the excess square under the X squared minus one will say we can concern this warm with this one here And then we have left with only the one of us. Cratchit on the X squared minus one.

In the given question we have to find derivative world. Uh, fax equals to one divided by access. Where so to end. The way you do with the help of limits, we have the formula that is LTD tends to zero and for that blessed -1 profits divided by yet. So here I am providing all the inputs. They're not eliminated, tends to zero F of explosives means one divided by explosives. Holy Square -1 of bone access where okay, no, just I am simply taking L. C. M. L. C M. Here should be explicit. Holy Square into success world. And after that access were minus Thanks Bella Search. Holy squid Now I am going to open the Holy Square. I am not opening the Holy Spirit of Denominated. I am opening the whole square of numerator. After that I should get access square plus city square plus do exit. Okay, now it is clear that our access square goes cancel out. And from the remaining terms I am going to take common eight at and if I take common age, then there remains minus it minus two words divided by explosive Holy Square into access Where now we know that we can cancel this edge are not for cancellation. I'm going to substitute. The limit means that tends to zero After the target zero -2. works divided by Ex last zero square. When to access square. No, this Can cancel your one x. And after that your final answer comes out. That is minor, stood divided by. Excuse me. Okay, So this is the value of derivative -2 divided by X.


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