### Stony Brook MAT 125 Spring 2015 Lecture 14: Derivative of inverse trig & logarithmic differentiation March 25, 2015

 Start what will be on the exam you need to be able to take the derivative of stuf you need to know how to power rule we'll right these down the power rule is the derivative of something like x to the 10th of 5x to the 10th so when you have a polynomial you do the derivative of all those terms so thats really easy people dont struggle wih power rule to much 0:30 product rule quotient rule chain rule chain rule is where it starts to get really annoying we just did the webassign on these implicite diffraction 1:06 be able to find the derivative of exponentials functions all logarithms 1:30 inverse trig functions guess what were doing today, inverse trig functions you also should be able to do something called logarithmic diffraction which sounds very fancy but its actually really easy its easy for me so thats kind of the list you need to be able to take the derivative of anything and at the end of next week in theory you should be able to take the derivative of anything' you might get in wrong but at least in theory you know what to do 2:01 and the rest of the course is going to be applying how calculus tools problems that i know youve been dying to do you need to know all those types of things we will be review that, we have a review session we have a review monday, we have stuff i have to put up lets figure out how to do the derivative of inverse 2:33 sin, why because theres a reason if you have y= inverse sin of x that can also be written as arc sin of x feel free to use either notation so we either write arc sin or inverse sin 3:01 that means that x is the sin of y what does it mean when we say x is the sin of y, so we draw a triangle and you have some angle y remember soh, cah toa opposite over hypothenuse so that means that the sin of y is x or x/1 3:33 and you can use the pythagorean theorem defined this side and get square root of 1-x squared so thats your triangle so far so good now were going to figure out how to find the derivative of arc sin so y is the src sin of x inverse sin of x, which means arc sin of y i take the derivative you have one, because x is 1 and this is cosin of y 4:02 dy/dx because we have to do it implicitly to get the derivative of y now lets isolate dy/dx derivative of 1 over the cos of y is dy/dx why did i draw this triangle because what is the cos of y remember soh, cah, toa cos of y is the adjacent 4:31 over hypothenuses adjacent is the squared root of 1-x squared the hypothenuse is just 1 so this says that dy/dx 1/square root of 1-x squared that is the derivative of inverse sin y= derivative of sinx 5:00 dy/dx 1 over square root of 10x squared that wasnt so bad you dont have to prove it you just have to do it so this is what you memorize my job is to explain where it comes from, your job is to know how to do it, you should also know where it comes from of course so what if i wanted to find 5:37 the derivative of sin pi x we got over formula 1/ squared root of 1-x squared 6:03 so here it would be 1/ the square root of 1 minus pix squared, dont forget to squared both of them why pi x because its this squared times the derivative of pi x because of the chain rule, this is pi squared root of 1 minus pi x squared 6:31 or pix squared x squared you do not have to simplify it by the way that reminds me, on the exam youll have so messy problems you dont have to simplify if you're doing product rule and quotient rule sometimes its handy to simplify but if we just say whats the derivative you dont have to simplify but if we say the second derivative youre better off simplifying the first and second 7:00 so the question is can you just leave it in this form yea absolutely we make that annoying by asking for the second derivative but of course whats the derivative of pi is it 0 are you sure? really what about pi squared you sure, its a number pi is just a number whats the derivative of pi to the 10th sill 0, how out pi to the x 7:32 we did that on monday since everyone was here on monday i dont need to do it again 8:11 lets do inverse tan so we said x is inverse tan of y, also known arctan or y 8:36 so if x is tan inverse of y thats the same as saying y to tan of x did i write that backward, yes actually sorry y is tangent of x x is the tan of y, sorry about that 9:03 if x is the tangent of y we draw a triangle i should make you do this yourself but so we have some angle y the tangent is x so its x/1 you use your favorite theorem, the one you only memorized pythagorean theorem 9:40 x is tan y that means when we have a triangle and thats y that means that tangent is x/1 so lets take the derivative, derivative of x is 1 the derivative of tangent is secant squared y dy/dx, just like the last thing 10:02 divide and we get 1 over 10:38 oh i just forgot that its the squared root of 1+x squared dy/dx is 1 over secant squared of y and the secant of y is the square root of 1+x squared over 1 because its the reciprocal of cosine 11:01 so this would be 1 over square root of 1+x squared squared which is 1/1+x squared if y is tan inverse of x dy/dx 1/1+x squared 11:32 well the square and square root and they go away everythings positive so you dont need absolute value bars so if i said if i had that lets find the derivative of tan inverse of 4x+3 12:27 its 1 over 12:30 1 plus this thing squared times the derivative of 4x+3 which is 4 and of course you can move the 4 on top and write it like that you guys still confused here thats 1 thats x right pythagorean theorem square root of 1 plus x squared 13:00 so 1/secant squared y is the same thing as saying cosine squared of y cos of y is 1/ suqared root of 1+x squared, so you are squaring it you get 1/1 because theres no squared the inverse sin theres no squaring memorize those 2 they will be very handy were not going to do inverse cosine 13:31 inverse secant inverse cot you dont need those if anybody asks you in another class whats the derivative of the inverse cosine of x its exactly the same as inverse cosx which is -1 okay so theres nothing really to learn if they ask for inverse cot its the same as inverse tangent except thats -1 you can figure it out on your own were not going to do inverse sec you can always figure them out if you need to, you have the trick 14:08 lets try another practice problem 14:32 lets see if you guys can take the derivative of that notice youre going to need the product rule dy/dx alright we have 2 functions you either do the first times the derivative of the second plus the second times the derivative of the first or the other one, doesnt really matter so we leave the first one alone and now we do the derivative of the second the derivative of inverse sin 15:01 is 1/ the square root of 1-x squared so this is going to be 1/ square root of 1-2x squared times the derivative of 2x 2 plus pkay so this is productrule so this is just the first half no you reverse we do the inverse sin of 1 15:32 and we have to do the derivative of sin 3x the derivative of sin of 3x is cos of 3x times 3 you do not need to simplify tht you do not want to simplify that i wouldnt want to simplify that we would never ask you to do anything beyond that of plugging in a number 16:00 we do something fun at x=0 yea thats about it and you say what happens at x=0 solving that for 0 you cant do it you have to use a calculator or a computer howd we do on that one should we practice another one yes 16:46 okay do the derivative of that how did we do the derivative of e to the tan inverse of square root of x well we just leave it alone 17:01 its e to the tan inverse square root of x because when you do the derivative of e to the something that term always stays there then we have to multiply by the derivative of the power because of the chain rule so whats the derivative of tan inverse of x its 1 over 1 plus square root of x squared which is just x times the derivative of the square root of x 17:32 which we memorize is 1/2 to the square root of x okay do you remmeber that one ill put that down, thats one of the ones i keep telling you guys o memorize very handy you should know where it comes from of course if y is the square root of x the derivative is 1/ 2 square root of x 18:00 that comes from writing it y=x to the 1/2 we are taking the derivative if you dont have it memorized i suggest you do same with y= 1 over x the derivative -1 over x squared why do i tell you to remmeber these youre going to see them a lot and these are the kinds of things we ask you to simplify or set things equal to 0 the square root cancels so if you wanted to simplify this 18:32 not a terrible idea this would be 1/1+x times 1/2 square root of x and if you wanted to show off you could put these to terms together but you dont need to the question was 19:00 you are doing the chain rule okay this is the chain rule, the first step doing e to a function the derivative is e to the function, thats the derivative to the e part thats the first outter most function the second function you have is tan inverse of x that 1 over, 1+ the square root of x squared then you have to do the derivative of the square root of x which is 1/2 to the x so thats your three parts does that answer that question lets do something similar 19:58 lets take the derivative of that 20:04 how do we do derivative of natural log you guys know the derivative of natural log thats all we did on monday and everybody was here so lets see y is the natural log and the derivative is 1/x and you get y is the log of a function of x 20:35 and the derivative is you make a fraction with the function on the bottoma nd the derivative on top i hope you can see that thats the derivative of natural log we did that on monday and i know how much you pay attention here we put the function on the bottom 21:04 and whats the derivative of inverse sin 1 over the square root of this thingy squared times the derivative of 4x which is 4, you put the 4 where ever you want thats it, thats all you got to do now of course you can simplify that but you dont need to 21:32 that simplifies to 4 over sin inverse 4x square root of 1-4x squared but you certainly dont need to do that you can just leave it there that is chain rule so once agin the derivative of natural log, the function on the bottom derivative on top, the derivative of inverse sin 4x 1/ square root of 1-4x 22:00 quantity squared times the derivatie of 4x which is 4, you also put the 4 up here you put the 4 above the fraction right there it doesnt matter did you get that one one more thing to learn well do some more practice 22:45 why is the radical here remember you said, function on the bottom derivative on top that should say the derivative of the bottom okay its one over the function ties the derivative of the function this goes in the numerator 23:02 however since that is a fraction if you want to do the algebra you can make it like this which is much easier the 4 would also be up here like that or it could be there the reason that this is true this is over 1 so if you flip it multiply this is in the denominator with this this is up in the neminator 23:53 its confusing because remember the inverse sin has a radical for the whole thing inverse tan doesnt have a radical at all, you take the radical out 24:11 okay suppose instead of asking something fun like y=x squared what if i asked you to find the derivative y=x to the sinx you know you go to be kidding me who would want to do that you would 24:31 it is not bring the sinx in front and reduce it by 1 it would be way to easy so how would we do that any ideas people who took calculus before here is what you could do, take the log of both sides natural log of both sides 25:02 then we could use the rules of log bring the power in front and you get natural log of y is sinx log x this is called logarithmic diffraction take the log of both sides then use the rules of logs put sinx in front remember that log of a of b 25:30 is b times the log of a remember that rule, now you do so now you can take the derivative because this is calculus so whats the derivative of natural log of y, this is 1/y dy/dx implicite right the natural log of y is 1/y and now we have to use the product rule for the other side sinx derivative of log x which is 1/x 26:02 plus log x cosx not quite done now we need to find, dy/dx we have 1/y times dy/dx so multiply 1/y you get dy/dx equals y times six/1x or sinx/x 26:33 plus logxcosx and what is y, y is x to the sin x 27:01 so it repeats as i see puzzled faces is y=x to the sinx how do i know how to do this because i have a function raised to a function okay so i cant use the rules i had before take the natural log of both sides and use the power rule, the rule of logarithmic powers to bring the sinx in front so now i have this natural log of y is sinx times log x 27:31 take the derivative of both sides the derivative of lny is 1/y dy/dx and the derivative of (sinx)(lnx) is sinx times the derivative of log which is 1/x plus log x times the derivative of sinx which is cosx and now we do some rearranging, put the y on the other side and replace y with x to the sinx lets do another one of those 28:09 how about y equals x to the e to the x so we have e, y=x to the e to the x so if you see that you know you need to use logarithmic diff. take the log of both sides natural log of y natural log of x 28:31 to the e to the x things dont cancel its not that easy you can put the e to the x in front and you get log of y is e to the x lnx those e's and logs dont cancel i know you want them to because they would make life so great but they dont alright so i take the derivative of the left side so i get 1/y dy/dx 29:01 and now lets do the derivative of the other side you have e to the x natural log of log x is 1/x plus natural log of x times e to the x which is just e to the x so far so good alright now you just do rearranging technically this is correct if you want to put the y on the other side, dy/dx 29:33 y times e to the x over x plus log x e to the x notice no canceling thats not the log of e to the x thats log of x times e to the x now you replace y with x to the e to the x 30:06 lets do one more make sure everyone has it oh if you forget to replace y, if you leave it like this i would not take off ill check with the TA's i think thats a correct answer but you should know thats nearly the correct answer but thats okay i dont think theres a big difference 30:30 but ill talk to professor sutherland to see alright lets give you one more of these to make sure you get the idea how bout y=x to the x you take the log of both sides log of y the log of x to the x bring the x in front 31:05 and you get y= xlnx 1/y dy/dx that left side is kind of boring the right side, product rule x times the derivative of log x which is 1/x plus 1 times the derivative of log x now we are going to simplify 31:34 so put the y on the other side and x times 1/x is 1 why would i have 1 because x times 1/x is 1 and replace the y dx to the x you get x to the x times 1 plus log x 32:04 i would put something like that on the exam, maybe i wouldnt do x to the x to the x im not writing the exam professor sutherland is other stuff you should be able to do 33:30 alright i give you f(x)=2x cubed +9x squared+12x-4 where does f of x have a horizontal tangent line what does that mean horizontal tangent line that means the tangent line is horizontal what does the slope mean when its horizontal 0, how do you find the slope of a tangent line take the derivative because what class is this calculus do you know anything else you know only one thing, you only got one thing in your tool box take the derivative, if you dont know what to do 34:02 take the derivative thats what youre here for lets do this, lets take the derivative find the horizontal tnagent line and set it to 0 thats an easy derivative its 6x squared +18x +12 so we can set that equal to 0 and you get x squared 34:32 +3x+2 equals 0 if you divide by 6 you couldve used the quadratic formula or if you were clear you would of divided everything by 6 first and throw the 6 out and this factors quite nicely x plus 1 times x plus 2 equals 0 so thats x=-1 x=-2 35:01 we could also ask you something like where is the function increasing and where is the function decreasing so what you would do make a number line and put on the 0's this would help us figure out where the derivative is positive positive means the function is increasing where the derivative is negative decreasing so you make a number line, put on the 0s 35:30 and now pick a value in each part of the number line pick a value less than -2 a value between -1 and -12 and a value greater than -1 plug it into the derivative and see if you got a positive of negative value remember we did this in precalc for graphing say we tried -3 -3+1=_2 -3+2=-1 a negative times a negative is a positive the function is positive there, that means its increasing and going up 36:02 increasing from negative infinity to -3 now you pick a number between -2 and -1, like -1/2 -1/2 +1=-1/2 thats negative -1/2+2=+1/2 so thats positive negative times a positive is a negative so its negative in this region so that means the function is going down there 36:31 decreasing from -2 to -1 so if you were graphing thi the grap is going up then its coming down so it has a maximum now its going down, what about the right of -1, how bout 0 0 you get 1 times which is positive positive is the function is going back up again so its also increasing 37:00 when the value is greater than -1, so we can ask you where is the horizontal tangent line where is the function increasing or decreasing, lets do another one one is never enough to practice 37:43 suppose we had that once again wheres the horizontal tangent line wheres the function increasing wheres the function decreasing take the derivative and you get 3x squared 38:02 times 6x -24 if this is a 6 point problem yoju earned yourself a couple of points now we set it equal to 0 we could factor that but if youre clever youll first divide it by 3 and then youll factor that 38:30 that factors into x-4 x+2=0 so you would get horizontal tangent lines at x=4 x=-2 so if the question says where does the function have a horizontal tangent line you stop at =4, x=-2 39:00 what if we said where is the function increasing and decreasing now you would use our number line test, this is called sign testing sign not sine pick a number less than -2 and test it in the derivative so like -3 pick a number between -2 and 4 like 0 pick a number greater than 4, 5 okay you are going to plug those into the derivative and see if you get a positive 39:33 or negative answer you take negative 3 you can plug it in anywhere in the derivative but its easiest when youre here -3-4=-7 -3+2=-1 a negative times a negative will give you a positive the function is increasing there pick a number between -2 and 4 i have 0 plug in the derivative -2 times 2 is a negative number 40:03 so your function is negative there then pick a value greater than 4 like 5 pick any number you want plug it in, you get a positive times a positive do its positive here so the function is increasing either when x is less than -2 or when x is greater than 4 40:30 and the function is decreasing between negative 2 and 4 notice gonna use interval notation you do not use square brackets it is not dreasing or increasing at an end point its increasing or decreasing in the middle lets do one more of these, lets make one slightly more annoying 41:17 suppose you had f of x=xsinx can you do that one, hang on that might be to hard 41:31 to hard without a calculator no you can do it thats to ahrd you cant do that without a caculator 42:05 lets take the derivative the derivative is e to the x times the derivative of sin which is cos plus the derivative e to the x, which is e to the x sinx thats a product rule right, so its e to the x times sin 42:30 plus sinx times derivative of e to the x which is e to the x set that equal to 0 and notice you can subtract your e to the x out and you get e to the x times cos x plus sinx so theres two possibiulities either e to the x =0 or cosx +sinx equals 0 43:00 where is e to the x equal to 0 e to the x is never 0 its always positive its never 0 just pay no attention how often is it 0 never good alright firgure the graph cosx+sinx=0 where sinx is negative of cosx of course you know this from the unit circle 43:32 or if you were to divided 2 from cosx you would get tangent of x is -1, you can solve it either way where is sinx equal to cosx 45 degrees or pi/4 its equal to negative cosinx at 135 degrees fourth quadrant ASTC, we all know what that stands for right 44:00 all students take crap its going to be in the second quadrant and the 4th quadrant its either going to be here or here so pi/4 in the second quadrant is 3pi/4 and in the fourth quadrant is 7pi/4 44:30 you should not write the answer in degrees i think we maybe would not take off but in calc everything is in radians nothing is in degrees alright ill see some of you on sunday, ill see the rest of you on monday