find-the-derivatives-of-the-given-functions-u4-sqrtln-2-te2-t-3

Question

Find the derivatives of the given functions.
$$u=4 \sqrt{\ln 2 t+e^{2 t}}$$

Tyler Moulton · Accepted Answer

we want to find the derivative. Dy dx for the function. Why is it with four times the square root? Natural longer than two X. Plus either two X. To do. So we&#x27;re going to rely on during of shortcuts we picked up from single variable. Help specifically rule 0 to 3 here on the left rule zero is D D X. B. To the U S B D U L N B D X D X E D U E D U D U D X. Rules +123 of the powerful product rule in general respectively. Let&#x27;s first rewrite. Why? So it&#x27;s more easily differentiable. Mhm. Why is equal to four times Ellen to expose either two X reached the power of one half. Now we see how the power rule chain rule Come in handy along with Real zero. So remember that the derivative of the natural algorithm of you is do you over you? So we have about the product rule. Do I. D X equals four times one half Ln two X. Let&#x27;s eat of the two X negative one half times the river. What&#x27;s inside to over to express to you? The two X times two. Don&#x27;t forget the term of two on the eve of the two weeks ending because of the chain rule. Thus our final solution is two Plus four x. either two x over X square root of natural algorithm of two X. Let&#x27;s either the two X or as highlighted now as a final solution.

Nicholas Barvinok · Answer

We want to find the derivative of the function square root of tea over T squared plus four. So we have a composition of functions here. So we&#x27;re going to use the chain rule and was our inter function is t over T squared plus four. Our outer function is taking an input and taking the square root of it or raising it&#x27;s the power of 1/2. So by the chain roll, we take the derivative of the outer function first, brace the power of 1/2. So, by the power rule, we bring down the 1/2 keep the input the same and subtract one from the original exponents. And now we have to multiply by the derivative of the inter function T over T squared plus four. For this, we&#x27;ll need the quotient. Rule the quotient rule for derivative states that yes, over g prime is g of prime minus f g prime all over g squared. So in our case was separate this In our case, we have the bottom is t squared plus four times the derivative of the top. The derivative of tea is just one minus the top tee times the derivative of the bottom. The derivative T squared is to t and the derivative of 40 because the constant then all over t squared plus four squared. And we see that maybe we can simplify this numerator a bit because T squared plus four minus two t squared is really minus t squared plus four. And so we could rewrite this thing perhaps a little bit nicer as one over two times the square root of tea over T squared plus four multiplied by minus T squared plus four over t squared Plus four squared and we&#x27;re done.

Amy Jiang · Answer

did you ask to take the derivative of the full term? So we see that we have a radical here. So let&#x27;s start there. Be ready this into an exponent so that when we pick our derivative can use are powerful. That&#x27;s t depart 1/2 and I&#x27;ll take are derivative. So that&#x27;s s Private E that&#x27;s people to will take this as winter. And this is another term and use our product, girl. So that&#x27;s four e t. Do you think about what happened? Plus the derivative of teaching about what house? That&#x27;s 1/2 Q to the power of negative 1/2 and then we are four et This could be canceled undisturbed into two. So yet or even a TV square root of tea, and then we have two or close to you to the tea and then over the square root of tea. Okay, this is our fallen derivative

Bobby Barnes · Answer

we want to find the derivative of the team is able to t squared minus one over T squared plus team minus two. So you could just start by using the caution war to take the derivative of this. But before we do that, let&#x27;s see if we can back this to possibly supply a little bit. So our numerator will too t plus one t minus one difference of squares and our denominator here. Well, we ask yourselves what number can multiply a negative to or what? Seven numbers won&#x27;t find negative too. But after one Principe&#x27;s t cost you over T minus one man, you might notice that he&#x27;s t minus ones or counts. And now we have t plus one over t plus two, which significantly simplifies what we&#x27;re trying to do here. And this one thing you need to make sure to write up onto the side that t cannot equal to what So I&#x27;ll take the drivel about t plus one of the teapots too. So, like I was saying, this is crucial. And so just to remind you of the suits so quotient rule. So the way I remember is the following. So ddx of some high function over some No function. So it&#x27;s slow. D Hi minus. Hi. Hello. All over the square of what&#x27;s below. So in this case, our highest people&#x27;s wanna Arlo is t plus two so we can Hey, t x t x there&#x27;s gonna bea t in this case dd t oh t cost one over t plus two. Well, this is going to be low. So maybe it&#x27;Ll be a good idea if I read these in different colors. Oh, all right. I like the new male Blue will be team plus one on the denominator green tea plus two so we can take low first. So t cross too. Team of what we have in the numerator t plus one minus what we have in the new mayor t plus one time to drive. They were respected T t too all over what we have in the denominator squared two plus two square. So the derivative of t plus one and keep us too Well, both of these are in the form out a derivative of t course from constant C, so we could go ahead and use some outstanding property to first break us. You team plus T c. The derivative of a constant is zero and a derivative of T to the first power will just be one. Since we could move the one out front using power rule and their team, you should try that. Get tea to zero and then teach zero is just what? So each of these derivatives here and become one So have I t plus two times one minus t plus one times one over t lost too square No. Now T minus t is going to counsel out and then two minus one is one So we can rewrite this as one over t plus two squared And then just recalling that this function should not be defied at T minus one and likewise are derivative. Should also be not to find that tea is equal to what So this is our derivative and t does not equal one is just one of the other values. This derivative is not supposed

Amy Jiang · Answer

Okay, So before I take the derivative, I&#x27;m gonna be right these two terms as negative exponents and then rewrite my medical as an exponents. So that&#x27;s 14 teach in a negative one plus 12 t to the night of four plus two actually weaken sleep like that since this is the constant. Okay, so now let&#x27;s take curative. So you have enough time of t is equal to negative one times 14 teaching a negative one minus one plus negative. Four times 12 teaching mega fuller minus one. And then the driver of the constant that secret to go. So we have negative 14 THC. The negative two minus 48 t to the negative five.

Alexander Wittmond · Answer

okay. If we want to find the door of the future of the first stop is to realize that a screw is the same thing is native E um Teet plus Ellen of two t to one half. Okay, Now we can use the changeable. So yet as prime of tea is equal to the derivative of one half, which is going to be one half acts with e negative T Ellen of two t plug it in, and that&#x27;s going to be one half one over each of the negativity. Eleanor. Tea to a negative one half, which is just one over square root of it. Don&#x27;t. Apple&#x27;s on of two tea. Okay. And then we have heat to the night of T plus Ellen of two t prime here. Okay, well, then, when we take the derivative of that here we have the same thing. Amen. We can just distribute the derivative over addition. Tough the addition sign and get negative e to the negative. Teague. Plus, And this is just going to be two times two tea or two over to tea on the twos. They&#x27;re going to cancel. Okay, Then you get one over T minus each of the negative t over two time&#x27;s a spirit of E to the negative T plus Alan of two tea. And if you want to simplify this, then we get one minus t e to the negative T. This is going to be over tea, and we can just go ahead and on simplified attraction as well. So you get it two t square of e to the negative T plus Ellen of two tea.

Problem

Find the derivatives of the given functions. $$y=…

Problem 23 Easy Difficulty

Find the derivatives of the given functions.
$$u=4 \sqrt{\ln 2 t+e^{2 t}}$$

Answer

$\frac{d u}{d t}=\frac{2\left(1+2 t e^{2 t}\right)}{t \sqrt{\ln 2 t+e^{2 t}}}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{d u}{d t}=\frac{2\left(1+2 t e^{2 t}\right)}{t \sqrt{\ln 2 t+e^{2 t}}}$

Basic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Grace H.

Kayleah T.

Caleb E.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus