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Diflerentiate the following iunctions. Use implicit. inverse diflerentiation . and logarithmic differentiation as needed [I0 points each](csex)xe (tan-!(3x))" ...

Question

Diflerentiate the following iunctions. Use implicit. inverse diflerentiation . and logarithmic differentiation as needed [I0 points each](csex)xe (tan-!(3x))" a.y = (Use logarithmic differentiation ) (sin x-2)(cos? x)b. In= sin-'(xly) _ 2c.y =V(ex?-2x-2 + sec(t) 9-xd.y = e-" sec-1 x ~ 3x6 + 2 cscx sin X 2xvBxey = In(tan(cot ~1 (Zx))) - Inx3-cos(8x) sin(8x)-x5

Diflerentiate the following iunctions. Use implicit. inverse diflerentiation . and logarithmic differentiation as needed [I0 points each] (csex)xe (tan-!(3x))" a.y = (Use logarithmic differentiation ) (sin x-2)(cos? x) b. In = sin-'(xly) _ 2 c.y = V(ex?-2x-2 + sec(t) 9-x d.y = e-" sec-1 x ~ 3x6 + 2 cscx sin X 2xvBx ey = In(tan(cot ~1 (Zx))) - In x3-cos(8x) sin(8x)-x5



Answers

Use implicit differentiation to find $d y / d x$ and then $d^{2} y / d x^{2} .$ Write the solutions in terms of $x$ and $y$ only.

$\sin y=x \cos y-2$

Again discussion of function of why it is given Synnex Cossacks 10 cube X divided by a road tax wrote checks. Can be written as expressed to the power Sorry wrote checks first of all there. Okay. And now we have to find out divide by dx using logarithmic differentiation. So we will you we will take natural logarithms in the heart of the question. So it will be L N Y is equal to L N and the same from abo Synnex cortex and cue backs divided by road tax. Okay? So L N Y is equals two. Okay, L N A don't be that is L N A plus L N B L N A by B. It will be an A minus Ln be So these three will be in plus sign and this will be the negative sign. Ok? So it will be Ln Synnex plus Ln course X plus L N 10-Q bags or we can say 10 x whole cube. Okay. And mhm minus Ln square root of x. Can be written and has access to the power one vital. So now L N Y. It will be Ellen Synnex plus Ln Cossacks. Okay so Ellen areas to the barbican Britain's B'elanna so it will be three Ln 10 X. Okay. And the same it will be won by two L N X. Now this is the question and we have to find out the divide by dx. So we will differentiate both are the question by X. So differentiation of L N Y. It will be won by Y tv by dx OK differentiation of Ireland Synnex first of all one by cynics and differentiation of cynics. So here it will be co sex. Okay plus differentiation of Ellen core sex It will be won by ko sex. Okay. And differentiation of cortex that is again minus sine X. Okay and plus three is already there. And differentiation of Ireland 10 X it is one divided by 10 X. And differentiation of 10 X. That is six square X. So it will be multiplied by six square x and minus one by two. And differentiation of island exit is one by X. Okay so now one by Y divide by dx. Ok it will be Cossacks by Synnex. Okay there test annex story. Cossacks by cynics that is cortex and minus Sine X by Cossacks. Their test annex and plus it will be three psyche square X by 10 10 X. Okay so psyche square is one by causes choir Okay and 10 X is caused by sign Ok so it will be one divided by Okay there was signed by here and caused by here so the same one bite connects that is Cortex Okay and psyche square X and minus one by two X. And now we will multiply both are the question by Y. So we will eliminate why from here. Okay. And multiply Y from my hair. Okay. And now we will put the value of why from the caution itself. Okay. That is Synnex Cossacks 10 cube X divided by the square root of X. And same from the about. That is cortex minus 10 X plus three court xx square x minus one by two weeks. Okay. You can also put the values of cortex 10 and 10 X in the tongues of sin accent. See you next time. Core sex. Okay but this can be your final answer of discussion. Thank you.

So we want to use long, different stations. So let's start by taking the natural log on both sides. So here we have the product of science coastline of tangents to accumulate this of the natural log of sign of X plus the natural log of coastline Durex. And here we have tending Cube. So that's plus three. The natural log of tangents. Elex all over. Actually, that's minus the natural log of the cube roots or to square roots. That's one half Macha log of X. So this is equal to the natural. Why? So let's find preservative with the fallings who have y fine times one over Why some people to one over Sign Rex Times coastline Elex plus do a review of the natural log That's one over coast Interfax times over coastline That's minus sign Must three over changing of x times. Um, they're of attention, which is seeking where directs an A minus 1/2 time Jax. And now you want to tell for wife rhymes, so I'm gonna multiply both sides by y. And now let's note that why was this term here? Okay, so we got t y over. G X is equal to y, which is a sign of X coastline of X change into cube vex over the square root of X. And this is times coastline over sign, which is coach engines of X minus, changing two bucks plus three Seacon square of X over. All right. Actually, it's right that that's cool Change. Innovex minus 1/2 X. So this here is our derrota.

Okay, so we're gonna be using log arrhythmic um differentiation to find this derivative. The way that we do that is we take the natural log of both sides before we take the derivative. So we're gonna have the natural log of Y. Is equal to the natural log of um X times sine of X. And what's this? What this is going to allow us to do is take this Synnex out of this power or exponent and put it out front. And so this is going to be equal to sin X times. The natural log of X is equal to the natural log of Y. And now what we can do is we can take the derivative of both sides of this equation. And so the derivative of the left side is going to be one divided by why? Since we have the natural log of Y and then multiplied by Y. Prime on the right side. This is going to be a product rule derivative. So I'm going to have the derivative of the first term or the derivative of sine X. Which is co Synnex multiplied by the second term which is the natural log of X. And then we have plus the first term. So Synnex multiplied by the derivative of the second term, which is one divided by X. So we have cosine X times natural log of X plus Synnex divided by X. And then on this side we have one divided by Y times Y. Prime. So we're gonna want to multiply both sides by why to get rid of this one divided by Y term. And so we have Y. Prime is equal to imprint sees co Synnex natural log of X plus Synnex divided by X times Y. And now all we have to do is go back up and look at what why was equal to why was equal to X. Time or X. To these Synnex power. So we go ahead and plug that in. For why we have Y prime here is equal to cosign X times the natural log of X plus Synnex, divided by X, and then multiplied by X. To the cynics power.


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