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Use logarithmic differentiation to find(sin(6x)) Ax...

Question

Use logarithmic differentiation to find(sin(6x)) Ax

Use logarithmic differentiation to find (sin(6x)) Ax



Answers

Use logarithmic differentiation to find the derivative. $$f(x)=x^{\sin x}$$

Todo algorithmic differentiation. So let's take a log of both sides by longer than rules that power can be brought into the front. So sign of X times Not for long, Alex. And now let's do implicit differentiation. So we end up with one over White Chainarong times. Why prime When we take the root of this break the X on both sides. Here we have product. We'll end up with coastline of X Natural log of X and then plants Sina Rex Over and Alan Scene. We can solve her life. We'll end up with x co sign X natural log of X plus sign of X all over X and times insulting by X sign of X when we multiply the Y on both sides of the equation.

They want to do lager ethnic differentiation again. Take in the natural. Get this Now let's use implicit differentiation. We'll get why prime over white is equal to Well, we have to do the product room so derivative Rex is just one and then we leave here alone. But now I do the opposite. Then we need chain role for the second factor is derivative one over that input signing times that heard of that input which is co sign So we end up with y Prime is equal to line of X to the multiplied by natural log of signing class X Times co tension of it.

The function of Alexis is able to sign a rex to the expiring and we're gonna perform logarithmic differentiations. Let's take full of natural law Global Thing. So I'll get extends the natural long of sign of X. Now implicit differentiation will have one of the right time is white friends is equal to well, this one is going to need the product. Take the derivative of the first factor. Leave natural Lagos I next one and do the opposite the excellent Take the derivative. Let's see, we have a general log to the input That's one over the input, sometimes the derivative of the input, which is co signer Rex. So that scene, what we can do is most of libel sides of this spiral wine, because what we have ISS the natural log of sign of ants. Plus, that's the X co tension of X. All of that times my original function sign of X to the X Oh,

So for this problem we have is right equals X to the sign of acts. And whenever we're dealing with these kinds of functions where we have an exponents with the X and the base with the ex, we want to implement logarithms Andi specifically the natural log. Here's why If we take the natural log of both sides, we can simplify this by getting rid of the exponents. So now we have a sine X out in front. Then we can take the derivative of this and perform implicit differentiation. So when we take the derivative of this portion right here, we get one over. Why times why prime? And that's going to be equal Thio here. What? You will use what's known as the product rule. You may have already been doing it a lot a tous point. So if the product role we take the first portion of the product and just keep it there but we multiply it by the derivative of the second portion. So this portion right here, natural log of X, give us one over X and then what we have is plus, we keep the natural log of access time and then multiply it by the derivative of Sine X, which is just Code Synnex. Now that we've done this, we want Thio, Combine all these terms and simplify further. So we'll get the sign of X over X and then this will just become natural log of X code Synnex Um And then lastly, what we want to do is multiply the Y on both sides. So when we do that, we have Why over here? But we know that why is ultimately equal to X to the cynics. So that's gonna be our final answer. Notice how we make things a little bit more messy by putting the natural logs in. But ultimately it made things simpler because it allowed us to do product rule rather than having to deal with the exponents, which are much more difficult to differentiate. Um, so it's a helpful technique that you will use oftentimes, whenever you see an exponents with a variable in both the base and the exponents portion


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