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Solve the given problems by finding the appropriate derivatives.The length $l$ of a rectangular microprocessor chip is 2 mm longer than its width $w$. Find the derivative of the length of the diagonal $D$ with respect to $w$

$$\frac{2(w+1)}{\left(2 w^{2}+4 w+4\right)^{1 / 2}}$$

Calculus 1 / AB

Chapter 23

The Derivative

Section 7

The Derivative of a Power of a Function

Derivatives

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let's find the derivative of this function G of W. We're going to be using the quotient rule to find us. So that is, we'll start out by just applying the rule g prime of W. The derivative egy of W is going to be the bottom term. So that is the squared of W minus w times the derivative of the top term. So 1/2 w to the negative half power plus just one minus. And then I'm actually going to move all of these terms down a bit because this is proving to be a rather long equation. So then minus the top term, which is, of course, the squared of W plus W times the derivative of the bottom term 1/2 w to the negative 1/2 power minus one, all divided by the bottom term squared square root w minus w squared. This is a lot. So let's simplify it at least a little bit to make it look nicer. So this will simplify to distributing out on the top. We have this well, 1/2 remains, then w to the negative half is one over the square root of w. So that will cancel with this when they're multiplied together, leaving us with just 1/2. So that's multiplying the first terms together. Then we have plus the squared of W minus 1/2 W over the squared of W um, which will actually simplify to just the squared of W on the top of the fraction and then finally, multiple in the last terms together minus W. Then we apply distribution to the other set of parentheses right here. This will give us once again the W's cancel. So just minus 1/2. Remember, this negative here is very important. Then the negatives cancel here to give us plus the squared of W. Then we have minus 1/2. Once again, it comes out to a squared of W on top won't supplying to these two terms together, and finally the negatives will cancel again to give us a plus square. W. And I just realized this should be a square root as well. So now that we have all of this established, let's move on to the bottom of the fraction. Squaring this out will give us squared of W. Times Square to W is W. Then we have minus two W Root W, which simplifies down to W to the three halves power. And then we have minus W squared. Of course, the negatives will cancel on the two W's to give plus w Square. Wow, this is a lot, so let's simplify it even further. 1/2 cancels with negative 1/2. That's good. Ah, we have this negative root w cancels with a positive route W. And unfortunately, the other terms aren't going to cancel. So we now have what we can combine these two route W's to give us to root W minus. So there's minus half through W minus half for W so minus a whole root w. And those will cancel in the future, as you can see, divided by W squared plus W minus two w to the three halves power. All right, we're almost done. Now we can cancel on the top, giving us just w to the 1/2 power divided by W squared plus W minus two w to the three halves power. And while we're here, we're going to factor out a squared of w from every term on the bottom. This will give us W to the 1/2 times W to the three halves plus W to the 1/2 minus to W to the two halves, which is just to w all right. And now we can cancel that on top and bottom to give us just one over W to the three halves plus W to the 1/2 minus two w. We can't simplify this any further. It's finally as simple as it can get. So after quite a lot of work, we've gotten our final answer.

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