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Pan I;Driwtiwssynlli below) Using Mathcmalicu: 141/ /problems 14,16.18.22,24,28,30,34 (seeThere un' several #ays uke derivative uine MathcmaticaConsider f6)-(1...

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Pan I;Driwtiwssynlli below) Using Mathcmalicu: 141/ /problems 14,16.18.22,24,28,30,34 (seeThere un' several #ays uke derivative uine MathcmaticaConsider f6)-(1-70)"Nay 1:D[(1 - 7.0)^6 .t] ~42(1 7t)5Raydefine the function first; then Use thc "prime' nolationnt_J:-(1-7*0)^6 f'[t] ~42(1 - 7t)define the function first, then use the "D" notation {[t_:=(1-7*t)^6 DU [t],t]-42(1 7t)5Defining function does help facilitate higher order derivatives: To take the third deri

Pan I;Driwtiws synlli below) Using Mathcmalicu: 141/ /problems 14,16.18.22,24,28,30,34 (see There un' several #ays uke derivative uine Mathcmatica Consider f6)-(1-70)" Nay 1: D[(1 - 7.0)^6 .t] ~42(1 7t)5 Ray define the function first; then Use thc "prime' nolation nt_J:-(1-7*0)^6 f'[t] ~42(1 - 7t) define the function first, then use the "D" notation {[t_:=(1-7*t)^6 DU [t],t] -42(1 7t)5 Defining function does help facilitate higher order derivatives: To take the third derivative of using the previously defined function f: Hey Hay



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Use Version 2 of the Chain Rule to calculate the derivatives of the following composite functions. $$y=e^{\tan t}$$

So I'm doing derivatives here. A age of X equals two X plus three times X plus four. Have to use the product rule here, derivative the first times the second plus derivative of the second times the first. No, I got to simplify that. Two X plus eight plus two X squared plus three X. Rearranging and combining like terms three X plus two X is five X. Something's not right here. They back up driven out of the second times the first river. The second was one so that acts shouldn't have been there. Okay, let's try it again. Sorry about that. The derivative of the second is just one. So now we simplify two X plus eight plus two X plus three. That's going to give me four X plus 11. Okay, be F of X equals two X cubed plus five X squared minus four X minus 3.75 So f prime of X would just be three times two is six. 10 minus four. That was an easier one. C. S equals t squared times t squared minus two T. Now we're going to use a D. S over DT which is live in its notation derivative. The first is to t 10 to the second. Using the product rule plus the derivative of the second, which is to t minus two times the first, which is t squared. Now simplify to t cubed minus four T squared plus to t cubed minus two T squared. So before t cubed minus 60 squared. All right. D candles. Okay. This one is going to be very neat. Do you Y over A D t five times one. Fifth is one X to the fourth three times one third is one extra squared two times one half is one X. That's it. E G of X equals five X squared to the fourth power. Now I wouldn't directly take the derivative of that. I would just call it five X. To the eighth power. Um And then g prime of X. Eight times five is 40 excess. Seventh R. F. S of T equals T. To the fifth power minus three T squared over to T. Such that T. Is greater than zero. So now we need to use the quotient role. Denominator is to t times the derivative of the numerator which would be five T to the fourth power minus six T minus derivative of the denominator is just too times the numerator which is T to the fifth minus three T squared to divide that by the denominator squared. So dana meter square is going to be four T squared. Okay now we've got to simplify that. I notice that I can factor a to t squared out of every turn and I'm just going to do that here so that's going to make this too. Get rid of that. Make this three two goes away that's going to be three and that goes away. Okay That left me with supposed to be a three up here. That left me with five T cubed minus six minus T cubed plus three. Yeah. Mhm. All over to five T cubed minus six minus T cube. Just just double checking. Right plus three. Okay that's going to give me five T cubed minus T cubed is four T cubed minus six plus three. Would be minus three over two. Which would be to t cubed minus three halves. Okay thank you for watching.

Of this over a want to use our cultural. Do you find the fallen vivid? So let's do that. So we have y popping that's equal to deserve it of of this term. So that's to t plus seven times are meaning, which is three t minus four and in plus conservative. This term, this is three times are remaining, Carrie. Okay, so let's expand this. So yet two times three, that's 60 minus a d plus 21 tea and then seven times for that's minus 28. And it will also have plus three t squared and then plus 21 t. Okay, so let's see. Um, we have a teeth word that's I have strong arrested beachy squared, and then or T, that's six it 2121. So we have six minus. It's plus 2121 heretical to 40 t and then we have, um, minus 28. Okay. And now, if reports, um, let me see if I could that correctly Shal that's six g. Sorry that you re a squared, so that's actually 90 squared. And instead of 40 we have negative change plus 2121 which is equal to 34. Okay, I know. Report Beat. It's expanders. So we have why that's equal to great. Cubed what is 42 squared? Plus 21 C squared and then plus negative. 20 or 28. So 21 minus four. That's 17. Okay. Mm. And now let's take charge of it. So we get nine t squared, plus two times 17. That's 30. 34 T minus 28. So we feel about two turns, which are derivatives, using a creditable on this, expanding their equal to the same thing.

We want to take the 1st and 2nd derivatives of the function G of two is 22 squared minus one off square times 32 square. We never know what the first derivative function look like. So taking the second derivative will always require us to be ready and familiar with all types of derivative problems. So none of the derivatives, the rate of change of function. Some examples including the typical derivative of polynomial that's X G x x squared plus two X. We might have to use the product rule, the chain rule and so on. At first it looks like we would have to use both the product and chain rule with G. But we can actually rewrite G by multiplying the polynomial out. But in the 2012 +226 minute, 12 to 4.32 square. Which allows us to use a typical polynomial derivative to solve. So you take the first derivative and then we'll take the second derivative from the first. So our product is or rather solution from the top to bottom is here. Our first derivative simply becomes 72 to the fifth minus 48 to 16, taking the typical rule, as was shown here. Then our second derivative is simply 3362 to the fourth minus 140 40 square plus six.

So for the first part, we need h crime of three, which is going to be given by F crime G of three multiplied with G prime of feet and G of three from the table is given us one. So we need to find f prime of one and multiply that with G prime of three f Prime of one is five and do prime of three is 20. So our final answer is 100. Yeah. For the second part, H. Pryor of two will be given by of prime give to into G prime too. And from the table, we see that GF too is equal to five and now we just need to multiply. That would g programming too. So if prime five is given as minus 10 and G prime of two was given this 10. So our final answer comes out to me and I give 100 for the third part. Where and we need to find Pierre for that is gonna be given by G prime efforts for multiplied with F prime for and from the table. Once again, effort for is given as one, and we just need to multiply that with private floor. Sergey, Prime of one is to and f prime of four is minus eight. Your final answer is my ass. 16 for part D, we need to find p prime of tool, which is going to be given by G Prime ever to multiplied with I'm trying to and f of two from the table is three. So we have the final three multiplied. With that, Brian was too g Prime of three is given us 20. And if crime of two was given us too. So our final answer is 40. For the last part, we need to find a judge Prime five. So we need f prod f prime Do your five and we multiply that with G prime of five. June 5 from the table has given us to on And she promised life as it is f crime of two from the table is too and g prime of to do you from the five. I'm sorry. This 20 sore final answer That's 40


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