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Points) Find the derivative of the functiony = (tan(x))_(tanx) ^x{log(tanx) +[xl(sinxlogx)]}...

Question

Points) Find the derivative of the functiony = (tan(x))_(tanx) ^x{log(tanx) +[xl(sinxlogx)]}

points) Find the derivative of the function y = (tan(x))_ (tanx) ^x{log(tanx) +[xl(sinxlogx)]}



Answers

Find the derivative of the following functions. $$y=\sec x \tan x$$

The first thing we want to do is to simplify a tangent co tangent. The simplification of tangent is sign of X overcoat side of X while be simplification of co tangent is coastline X over sine X? The next thing we want to do is to get our denominators the same. So on this side of multiplying top and bottom by side x no the other side doubles applying top and bottom by coastline X This will turn out to be sine squared x over co side X side X plus co sign square debts over cause I necks Scientifics. The next thing we want to do is to combine our fractions were using this property a over see plus minus B oversee is equal to a plus minus B over. See So this will like co side squared X closer Dsquared at us All of this over Costa decks Sigh next. Now we're going to use the identity of closer and squared X plus side squared X equal to one So this will look like one over close Idexx Sine X thinks of a no we are going to use our identity. Another eternity co side x Sorry X is equal Teoh sign of two X well supplied by 1/2. This will look like one over one Have signed up to acts This could also be reread knows one over side of two acts over to If we use our fraction Rule one Overbey oversee is equal to C over B. Our new fraction will look like to over signed of two acts. Now we're prompted to use another identity. One over side of X is equal to Cosi Contak this will be our fraction. Will know turn into to co sigint to axe. And after we have this, we will know find are derivative so it will look like D over DX of two. Cosi can't two x. The next step will be to take out the constant two D over DX of Cosi Ghent two x Now, To solve this derivative, we want to use the changed role de after you over. The X is equal to d f over g you well supplied by do you over detox. In this instance, R f is Cosi interview and are you is two x put together. It will look like to do you ever do you of Cosa Hosea, can't you? Well supplied by D over DX of two arcs. Now our next step is to find D'You and DX de over. Do you? Of course, Eakins, You is equal. Teoh Negative Coach. Religion. You co seeking to you now? We're looking for it. DX so D over DX of two X is equal to two. Now that we have our d u and r d x, we could put everything together two multiplied by negative Coty, Agents of you Costa can't of you well supplied by another two Replacing use with the actual values and simplifying everything Together you will get negative for Coach and Jude's of two x Cosi can't two acts.

So the trick atomic identity that I'm going to use in order to find the derivative of tangent of X. It's just that tan X is equal to sine of X divided by co sign of X. And so I'm just gonna go ahead and plug this in for 10 of X. So we have Y here is equal to sine of X, debated by co signing backs. And now when we find this derivative, it's just gonna be a quotient rule involving sine of X and cosine of X. So why prime is going to be equal to the derivative of sine of X or the numerator understanding across and rule here, which is equal to cosine of X divided or sorry, multiplied by cosine of X. Or the denominator. So they would be co sine squared of X. And then we have minus sine of X times the derivative of cosine of X, which is negative sine of X. So we're actually gonna get plus sine squared X. And then we have divided by co sine squared X. Or the denominator square. And so cosine squared plus sine squared is equal to once. We have one divided by coastline squared X, which is equal to see can't squared X.

This question asked us to find the derivative of the function Sign of Tan of two acts. What we know is that we're gonna be using the chin ruled to get wide prime. We know the derivative of Sign is co sign. We have co sign of 10 of two X. This is what's on the outside now it's do. It's on the inside. The derivative of Tan is seeking square, so seeking squared instead of acts we have two exes You can see the derivative of two acts is simply too. Therefore, this simplifies to two seeking squared to axe crow sign of 10 of AKs and remember, he goes out of 10 of two backs. And remember, the exact order of these doesn't really matter. As long as the coefficient isn't front, you could do co sign of tan before you do seek and square the exact

What question? 11, We got to differentiate this particular functions. So differentiation is not trabajo y dash definition of tennis sex square X standard formula on for extra square. We're going to use the power rules or to comes down and the power decreases by one or two minus one is just one. Since this cannot be simplified any further, so this will be our final answer.


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