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1. (6p.) Find the derivatives of the given functions Y=21- +4xb) y=(x 2x )6x2 - x - 3)1+2*3 C) y = ZVxY = 2-tanxy = secOtaneY =cOsI 2sinx...

Question

1. (6p.) Find the derivatives of the given functions Y=21- +4xb) y=(x 2x )6x2 - x - 3)1+2*3 C) y = ZVxY = 2-tanxy = secOtaneY =cOsI 2sinx

1. (6p.) Find the derivatives of the given functions Y=21- +4x b) y=(x 2x )6x2 - x - 3) 1+2*3 C) y = ZVx Y = 2-tanx y = secOtane Y =cOsI 2sinx



Answers

Find the derivatives of the given functions. $$y=\frac{2 \cos x^{2}}{3 x-1}$$

And this problem, we need to determine the second derivative of the given function. Why is equal to two X. 20 par seven minus actually power six minus three eggs. Now the first derivative will be two times the derivative of X. To the power 7 to 7 X. To the power six minus six times. Actually power five minus three times the derivative of X, which is one. So this will be 14 X. To the power six minus six X. To the power five minus three. Actually determined the second derivative. That would be the derivative of this expression over here. So that will be equal to 14 times the derivative of X. To the power six which is six X. To the power five minus six times the derivative of X. To the power five which is five X. To the power four minus the derivative of three which is a constant and thus the derivative will be zero. So 14 times six gives 84. So this is 84 X. to the power five minus six times five, which is 30 X to the power four has. The second derivative of the function will be 80 for extra power five minus 30 extra power for.

We want to find you I. D. X. For the function why equals cosine of one minus two X. So to do. So we're going to rely on derivative shortcuts. We've picked up two single variable calculus such as those that you're gonna metric derivatives especially that's the most important rules are listed below. On the left rule zero is D D X and X equals Cossacks and DDX. Cossacks making Synnex. We also have rules one through three listed the power rule, product rule in chain rule. And this particular problem will rely on rules zero, three and 1 to solve. So we put this in the easiest form of differentiate that's already the form is currently in. So why would anyone minus two X. Next we proceed to solve. So let's differentiate. This gives dy dx equals negative sign one minus two X. The derivative coasts times negative to the derivative of what's inside the the chain and power rules So our negatives cancel. And our final solution is Dy dx is equal to two times a sign Of 1 -2. X.

All right. We want to find do I. D X for Y is equal to two times a sign of two X cubed minus one to do so we're going to rely on derivative shortcuts that we picked up through the single variable calculus. Those shortcuts are listed here on the left. Most relevant rule is rule zero. D D X in X equals Cossacks and DDX Cossacks from negative sine X. Also on the right are rules one through three of the power will product rule in the chain rule. We'll have to use rule 03 and one together to solve this problem. So we want to write this in the form. That's easy. Easy to differentiate why is already in the correct form. So you can receive the derivative. So derivative gives us dy dx equals to coast to execute minus one. This is taking derivative of sine times the derivative of what's inside in six X squared note that we obtained six X squared from the power rule for two, a few minus one. Thus our final solution is 12 X squared the coastline of two X cubed minus one.

We want to find the derivative Dy dx for the function Y equals to Costa can three X over excuse to do so, we're going to rely on the right of the shortcuts. We picked up a single variable cap in particular. We're going to make use of our recently learned trigonometry derivatives as such. I've listed the major trigger magic through it is here for all six string functions. We need to use the coast. You can't derivative, which is negative coast, you can't cook tangent. I also made note of the chain product rule and so on. We'll need both the chain rule and product rule to solve this problem so we can first rewrite why? As to Costa Can three X. M. X mega second. This allows us to start off the derivative with the product rule, thus proceeding to differentiation. We have dy dx equals negative to Costa three X. Co chanted three X times three Xn 92nd. It supplies the coast can't derivative in the chain rule on the left term minus two. Costa can't three X times X. Negative three times to supplies derivation or differentiation too expensive to thus combining like denominator, we have negative to Costa. Can't three X times three X. Co, changing three experts to all over execute.


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