find-the-derivatives-of-the-given-functions-y6-tan-ex1-3

Question

Find the derivatives of the given functions.
$$y=6 	an e^{x+1}$$

Tyler Moulton · Accepted Answer

Yeah, we want to find the derivative. Dy dx for the function Y equals six tangent of E. D X plus one to do. So we&#x27;re gonna rely on during a short cuts that we&#x27;ve been picking up throughout single variable calculus. Specifically Rule 0 to 3 here on the left. Real zero states. DDX BTU is B T u L n B D u D X D X. Interview is E d u d u D X rules one through through the power rule products and chain rules, respectively. We also rely on the fact the derivative of tangent is you can&#x27;t square to solve so we can combine real zero rule three and the tangent derivative I just described to solve, we note that Y equals 60 X plus one is already in its most easily defensible forms. So dy dx is equal to six. You can&#x27;t squared and E X plus one by the chain rule, the derivative E X plus one is still E. To the X plus one. So our final solution is six E. To the X plus one time U. C. Can&#x27;t square of E to the X plus one.

Theresa Nguyen · Answer

we&#x27;re looking for the derivative of Why is equal Teoh Negative X cubed coastline of X plus three X squared sedative X plus six x co. Sign of eggs minus six sign of x To solve for this, we&#x27;re going to use thes sub difference rule to break down our derivative Be some difference. Rule is F plus minus G prime is equal Teoh f prime plus minus cheaper Applying this rule are derivative Will look like minus D over dx of x cute co sign X plus de over T x three X squared Sign of ass plus D over DX of six co sign of X put minus de over D X of six Sign. Relax. So now we went to start solving for all of these derivatives We&#x27;ve broken down starting with D over DX as ex cute co sign X to solve For this derivative, we would to use the product rule f times G prime is equal to ask time terms Jeep plus f times g prime here are s is equal to x huge So our f prime will be three x squared RG is coastline of X so our G prime will be negative side effects Putting it all together We will get three X squared co sign that X plus negative sign of X Well supplied by X Cute simplified that will look like three X squared co sign of X minus X cubed site of X This is one derivative down. The next derivative will be D over DX of three X squared site of acts to find this derivative were also using the product rule here R f is s squared so our f prime will be to S R G is side of X so I keep time will be co side of X putting use. Together we will get three multiplied by two x sign of X plus co sign of X terms X squared. Our next derivative will be D over DX of six x co side of s. To do this, we use the product rule again. So our f is X last time will be one RG is co side of X So our g prime will be negative side of X Putting these together we will get six times one times coastline X close, negative sign Look x Times X. Now we&#x27;re going to put all of the derivatives we&#x27;ve taken together, which will be okay. Negative. Three X Squared Co. Sign of X minus X cute side of X plus three times two x sign of X Plus Co sign the X Times X Squared plus six co. Side of X minus six. Sign of X. Let&#x27;s just take this down. Co spin of X minus six Sign of X minus six co side of X. If we take all of these and we simplify, we will get X huge sign of X.

Todd Vawdrey · Answer

where this problem we&#x27;re finding the derivative of negative six extra. The fourth sign. Six sects. For this, we&#x27;re gonna need to do the product rule. So I have two functions being multiplied together. We have the negative six x to the fourth, and then we have the sign of six X. So for the product rule, we&#x27;re gonna go ahead and get that D y d x is equal to we take the derivative of the first, which is negative 24 x cubed times The second sign six x plus the first thing negative six x to the fourth times, the derivative of the second, which is the co sign of six X. And then we multiply that whole thing by six because of the chain rule and the derivative of six exes. Six. So then we can go ahead and simplify this thing. So you get negative. 24 X cube signed six x. I&#x27;m gonna put the sixes together minus 36 extra. The fourth co sign of six X. You may also see this factored so they have a common factor of negative 12 x cubed. And then what would be left if I kept that is signed to the six X Plus three ex co sign six x So you may also see it like that, depending on your teacher, but they&#x27;re both correct.

Theresa Nguyen · Answer

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&#x27;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&#x27;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&#x27;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&#x27;t you? Well supplied by D over DX of two arcs. Now our next step is to find D&#x27;You and DX de over. Do you? Of course, Eakins, You is equal. Teoh Negative Coach. Religion. You co seeking to you now? We&#x27;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&#x27;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&#x27;s of two x Cosi can&#x27;t two acts.

Theresa Nguyen · Answer

in order to find the derivative of why is equal to tangent w over one plus 200 w. We will first to use the quotient rule f over g prime is equal to F prime times She minus g prime times f All of this over g squared. The application of this rule will look like the over d w a tangent w well supplied by one plus tangent of you minus d over d W of one plus two ancient W multiplied by seamanship. W all of this eyes over one plus tangents w squared. The first thing that would to dio is to take the derivative of tangent w de over d W of tensions. W is equal to seek int squared w Next we&#x27;re looking for the derivative of one plus tinge of w de over D W of one plus tangent. W is equal to seek and squared w. So now that we have these values, let&#x27;s put them back into the quotient role. Now we will get sequins Square W well supplied by one plus Children W minus Seacon Square W well supplied by tangent w All of this over one plus tangent w square. The next thing we want to do is to simplify our tangents and seek its into science and co signs. We will get one plus signed w over close i w multiplied by one over co sign W squared minus one over co sign of W squared Well supplied by sign W over co sign w. All of this is over one plus sign w over co sign w squared so we can actually even simplify further. Let&#x27;s look at one plus side of W over coastline w We could simplify this to be off gases squared. We could simplify it further to look like co sign w less sign w squared over co sine squared w So our new function will look like one over co sign w squared well supplied by sign, don&#x27;t you? Over coast and w plus one minus one over coast I w squared well supplied by sign of w over close and w All of this is over. What we just simplified earlier co sign w plus signed up. You squared over co sign squared. Next we will simplify one plus signed out to you over Costa W well supplied by one over closely w squared. This will be co sign of W plus side of W over Cosan cubed W. And then we will also now simplify one over co sign W Squared won&#x27;t supplied by Sign W over coastline W That gives us a sign. W over co side cute w With these values simplified, let&#x27;s put it back together. You will get co side W plus sign W over co sign Huge W minus signed W over Coast Side. Cute w All of this is over. Close eye in W plus signed up. You squared over close side squared W. The next thing we&#x27;re going to do is to combine our fractions. It will look like so. Coastline W plus sign W over coast side cubed W minus sign W over co sign Cubes W, which will be equal to co sign w Quest. Signed w itis scientific view over co sign Cute W. And if he simplify our combined a fraction, we will get co sign w over co sign Cubed W Now if we take out, he cross out the coast, i w we will get one over co sine squared of you Now let&#x27;s put it back into the original problem. One over co sine squared W over co sign of W plus sign of W squared over co sign squared W. The next thing we want to do is to divide our fractions. A over B over C over D is equal to a times D over B times C. This will look like close in squared W over coastline squared W well supplied by co sign W plus Scient of you and Square. The next step is to cancel the common factor, which is closer to W. So let&#x27;s cancel coastline, W coast and square W. And we will get one over. Co sign Debbie you plus sign W squared. The next thing you want to do is to expand people God&#x27;s one over co sine squared W plus two closely w Times Side w plus I sine squared w simple fired that will be one over co sign squared W plus sine squared w plus to co sign W. Times Side W. Now we&#x27;re going to use the I Don t of co sign squared of X plus. Sine squared of X is equal to one that gives us one over wood plus to co sign w signed W The last identity we&#x27;re going to use ISS to co sign w signed w is equal to sign of two w applied. It will look like one over one plus sorry of two w this is the final answer.

Alex Roush · Answer

Okay, so we have to find the derivative of this immediately. We see the product of two functions. So, product, it&#x27;ll kind of screams at us now. In the past, whenever we multiplied to polynomial is together, we actually went ahead and expanded it out and discovered that taking the derivative of that was usually easier. But because this is called a transcendental function, it&#x27;s an exponential. Combined with the polynomial, The product rule really is the best way to go. Okay, so f prime of X is going to be the first function times the derivative of the second. Okay, So since this is a polynomial, the product rule applies here. The power roll place. So I&#x27;m gonna take these exponents, bring them down his coefficients. Okay, so this would be three acts, and then I&#x27;m gonna decrease the exponent by a factor. One a three minus one is to okay. And then I&#x27;m gonna bring this sex, put it down as a coefficient. So I have three times 2 to 6 x to the first power. Okay. And this is linear. It&#x27;s looked at six. And then the derivative of a constant is zero. Now I take the second function, and I multiply all this times the first, the derivative of the first function. But the derivative of the exponential is just the exponential. It&#x27;s no what I&#x27;m gonna end up having if I simplify this little bit, I mean a factor in each of the X out. And I get what? Well, have one x cubed trim. Okay, I have this minus three acts in this positive three X that are gonna cancel. Have this negative six sex in a positive six sex, which are also gonna cancel. And I have a negative six and a positive six that also cancel. So everything cancels except the sex que term, which is really nice. So that&#x27;s your answer.

Problem

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

Problem 30 Easy Difficulty

Find the derivatives of the given functions.
$$y=6 \tan e^{x+1}$$

Answer

$\frac{d y}{d x}=6 c^{x+1} \sec ^{2} e^{x+1}$

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

Caleb E.

Kristen K.

Precalculus Review - Intro

Functions on the Real Line - Overview

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$\frac{d y}{d x}=6 c^{x+1} \sec ^{2} e^{x+1}$

Basic Technical Mathematics with Calculus

Grace H.

Anna Marie V.

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.

Anna Marie V.

Caleb E.

Kristen K.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus