### Stony Brook MAT 125 Spring 2015 Lecture 10: Derivatives of trigonmetric functions March 4, 2015

 Start were gonna do more derivativing today as just a reminder of what we did the other day we learned the power rule power rule is what you use with things like polynomials y equals something to a power with a constant in front is okay then the derivative 0:30 you just bring the power down and reduce the power by 1 that was very simple right, the derivative of x to the 10 if y is x to the 10 then the derivative is just 10 times x to the 9 if youre going to do the second derivative which youll learn this notation later is 90 1:00 x to the 8, and you can keep going till you run out of derivatives, youll run out of derivatives at 10 so we can ask for the 3rd the 4th the 5th the whatever derivative okay and each time you would bring the power in front so jus to heat this up, you have y=x to the 4 1st derivative is 4x to the 3 second derivative is 4 times 1:30 3x to the 2 3rd derivative is 4 times 3 times 2 does that look familiar you know what function that is, the fourth derivative 4 times 3 times 2 times 1 x to the 0, what is 4 times 3 times 2 times 1 4 factorial x to the 0 is just 1 the derivative of x to the 4th the 4th derivative of x to the 4th 2:01 is 4 factorial what do you think will happen with the 5th derivative youll get 5 factorial oh im sorry let me rephrase that the 5th derivative of this will be 0, because the 4th factorial will be constant so once your derivative passes that number you just get 0 again if y=x to the 5th then the 5th derivative will be 5 factorial 2:31 if y=x to the n then te nth derivative is n factorial for those of you who dont know what n factorial is is n times n minus 1 times n minus 2 all the way down to 1 so if you have n to the 100 100 times 99 times 98 97, all the way down to 1 okay very big number the next derivative after that is 0 so if i said whats the 100th derivative 3:02 x to the 99 its just 0 so far so good, thats the power rule you can have some fun with that look for patterns we normally ask for the first and second derivative, you can do lots with the first and second derivatives we dont really care beyond that unless we feel like just punishing you NHE 127 youll learn about something called paler series and thas where we use more derivatives 3:34 ah this is a notation thing so theres another way to write derivative notation dy/dx its called live denotation what you are doing is doing d of dx of y, which we shortened to dy dx so if you take the derivative of that its d/dx of dy 4:00 dx which is d squared of y dx squared thats why you use that notation the third derivative would be d3y and dx 3 and so on but you should get very comfortable live notation because youre going to use that much more than the prime notation thats just a quick short hand but a lot of things you really have to put it in this form to work with it 4:32 we all understand this but more important y=kx to the n y prime is knx to the n-1 then we did product rule product rule when you have two functions multiplied together when you have y is function f of x 5:02 times function g of x and the derivative is the first function times the derivative of the second function plus a second function times the derivative of the first function or the other way around becaus your doing multiplication and addition order doesnt matter so you can switch these two thinga 5:30 for example if you had f of g t cubed e to the t, oh wait getting ahead of ourselves t cubed times 5t times t squared now you can certainly just distribute that but if you wanted to do the prouct ruel 6:03 f prime of g is the first function times the derivative of the second which is 5 pkus 2t plus reverse derivative of t cubed is 3t squared times 5t plus t squared 6:37 then we learned derivative of e to the x which is e to the x that is very challenging but many of you will see to memorize that the derivative of e to the x is e to the x the second derivative is e to the x the third derivative is e to the x 7:03 its just e to the x isnt that handy? very good to know e to the x also, e to the x is never negative or 0 because you remember what the graph of e to the x looks like the graph of e to the x looks something like that so notice its never 0 and its never negative this will be very useful because remeber when i told you youll have e to the x 7:30 youll have an equation and find out where its 0 if e to the x is the equation, e to the x s never 0 so you can just take it out so where would that show up suppose you had f of x x cubed e to the x and you wanted to take the derivative notice its two functions so you would have to use the product ruel 8:00 this is something people trip up on all the time so the derivative is x cubed times the derivative of e to the x plus e to the x times 3x squared notice e to the x is in both of those terms so you can pull that out 8:33 in fact you can also take out an x squared if you wanted if you were setting this equal to 0 notice x squared can equal 0 at 0 and x+3 can equal 0 at -3 but e to the x is never going to be 0 cause e tot he x is always positive so if we had to find the 0s 9:00 you would just find where this is 0 this is 0 and ignore the e to the x we go that dont go it, what dont you get so far so good another word we learned the quotation rule 9:41 is you have f of x over g of x this one is lidehi-hidelo the derivative the low function times the derivative of the higher function minus the high function times the derivative of the lower function 10:00 over the lower function squared so for example 10:33 you want to find the derivative of that you take the derivative you use the quotient rule bottom function times the derivative of the top function which is 3x squared 3x squared plus 4 minus top function times the derivative of the bottom function 11:01 so x cubed plus 4x times 2x all over x squared minus 1 squared thats not so bad the messy part is when you have to do a second derivative because you have to simplify theses things, lots of times it simplifies quite nicely 11:32 lets practice this one you would multiply out, youd get 3x to the 4 plus 4x suqred minus 3x squared minus 4 minus 2x to the 4, minus 8x squared all over x squared minus 1 squared by the way when is a fraction equal to 0 A fraction is equal to zero When the numerator is equal to the 12:01 Provided the denominator is also not zero When you have to find zeros you only care when the top is zero What makes a fractions zero the bottom is another thing So you'd simplify that let's see You get x to then4 X squared-7 X squared, 7X-4 to the fourth Which is easier to work with over 12:34 X squared -1 squared That's quotient rule now let's learn something new 13:02 I am going to wait a few more seconds and then I'm going to the erase this 13:32 Okay we have to remember our trigonometry for a couple minutes Remember the unit circle Something like that I'm only going to draw a quarter of the circle Just the first quadrant you don't need the whole thing 14:03 There you go that came out better okay Let's call that o for the origin And remember when you do circles you have a radius that sticks outside from the origin I'm going to label some of these points call this point B And we're going to call this a where intercepts at the X axis 14:30 It's a unit circle unit meaning one the radius is one Which means this distances also 1 OP is one an OA is one How do we find the coordinates of point P We have the x-Cordinator in the y-coordinate so if you remember from soa-cah-toa 15:05 This distance is cosine theta because it's adjacent over one And this distance is the sign of theta Where is the tangent of theta Turns out what letter did I call that t This is the tangent of theta 15:32 thats why we call that tangent if you ever wondered why thats the tangent of a circle now you know, this makes a right angle and why is that the tangent well you think of this angle, this is theta tan theta is opposite over adjacent 16:04 which is ta/oa but oa is 1 so the tangent of theta is ta/1 so this length is the tangent of theta, you can think of the unit circle stuff thats the sin thats the cosin thats the tangent and if you think of this arc as that arc gets closer and closer to 90 degrees that line is gonna get higher and higher part of why the tangent goes up to infinity 16:32 in case you ever wondered one last thing lets draw that line ap pa depending on where you grew up alright sp notice a couple things the area of triangle opa 17:01 has to be less than the area of this whole sectore the sector opa which has to be less than the area of the triangle all agree so the triangle is less than the whole sector cause you have this shaded piece out here and the sector is less than the big triangle 17:36 well what is the area of the triangle opa well its 1/2 base time height 1/2. whats the base, the base is oa which is 1 and the height is sine theta see that in color 18:04 this triangle has a height of sine theta and a base of oa so far so good the area of the big triangle well if the base is 1 and its height is tan theta 18:32 how do we find the area of a sector, you guys remember how to find the area of a sector lets review that you have a circle, a pizza you want ot find the area of a slice of pizza its just proportions, if you know this radius 19:02 and you want to find the area of that slice the ratio of the angle to 360 but were doing this in radians so 2pi radian is the area of the piecd divided by the whole circle so again the angle compared to the whole thing equals the sector compared to the whole thing cancel pis 19:34 and you get 1/2 theta r squared equals the area or r squared theta, the area of that circle is 1/2 theta r squared or r squared theta theta is in redians and by the way in calculus everything is in radians were not going to do anything in degrees sorry folks thats why you learned radians last semester in trig 20:02 1/2sin theta is less than 1/2 r squared theta is less than 1/2 tan theta so far so good so we can cancel the 1/2 and you get sin theta is less than r squared theta is less than tan theta 20:39 lets divided everything by sin theta so this is 1 this is r squared oh im sorry well do it next set, r squared theta 21:00 over sine theta and what os tan theta. sin theta that simplifies sin/cos/sin lets see sin over cos divided by sin is sin over cos times 1/sin is 1/cos 21:30 okay so thats 1/cos by the way what is the radius of the circle you said this is the unit circle so the radius is 1 so we can take r squared and replace it with 1 were starting to get things good here 1 is less than theta over sin theta is less than 1/cos theta okay flip everything upside down and you get 22:03 1 is greater than sin theta over theta is greater than cos theta thats called the reciprocal okay thats a fun thing, alright what happens when theta goes to 0 is you want to do the limit now as theta goes to 0 22:30 1 is greater than this thing whats the cos of 0, 1 thats the squeeze theorem so that tells you that the limit, this is the goal the limit when theta goes to 0 sin theta over theta equals 1, providing we are doing things in radians so all of that was to make sure you know and understand this, thats enough theory i like to do 23:02 but i like to throw it in once in a while to keep everyone honest r squared 1 for 1 cause the radius is 1 so once again thats a proof the idea ia because were going to need this in a couple minutes to make sure that you understand that the limit as theta goes to 0, sin theta over theta is 1 we like to test this 23:31 so lets put that some place safe im going to swith it to x cause it doesnt rally matter very important limit to know make sure you do because we will test you in some variation of this a couple of other things that you might not know suppose we want to ind 24:02 the limit as x goes to 0 of 1-cisx/x what would you do well take some guesses throw out some numbers pi, 1 0, infinity, cant be determined why dont i wait for professor kahn you think its 1, who agrees its 1, who thinks its 0 why dont we multiply the top and the bottom of this 24:34 by 1+cosx what do you get then, okay so now the limit as x goes to 0 1-cos times 1+cos is the conjugate so its cos squared somewhere in the back of your head that should sound familiar 25:00 1-cos squared is sin squared oh what do we do with that, we can break that into 2 limits we can break that into limit x goes to 0 sin x over x times sinx 25:32 over 1+ cosinx the limit as x goes to 0, of sinx/x we just learned that equals to 1 so this is the limit as x goes to 0 is 1 whats sin of 0 0 and whats 1+cosx 2 because cos of 0 is 1 so that times 0 is equal to 0 anybody who agreed with me 26:02 got a 0, good thats our second q you guys are smart to wait were building blocks for learning stuff im going to now erase this 26:31 last thing, angle addition formula remember this from the regents that was on the sheet that you can break and use through the exam so you didnt really have to remember it sina cosb plus cosasinb remember that regents, algebra trig regents 27:00 howd we feel about that one we liked that one 27:35 why am i doing all of this because were gonna learn now what the sin of x is lets find the derivative of sin of x 28:00 were gonna learn what the derivative of sinx is then you can tear out the other pages in your book and say im just gonna memorize this derivative of sinx, lets do the limit as h goes to 0 sin of x plus h minus sin x all over h 28:39 we gonna do this as a team okay sin of x plus h thats this thing, thats gonna be sinx cosh 29:00 plus cosx sinh minus sinx all over h cause you use this formula, this formula, im not gonna tell you where this comes from its an mat 123 thing not a mat 125 thing im just going to do a tiny bit of rearranging, im gonna more the 2 and the first terms 29:35 and rewrite this now as limit as h goes too 0 of sinx sinx, cosh minus sinx plus cosx sinh over h, i didnt do anything i just flipped the second and third terms now lets break that into 2 limits 30:06 first two terms i do sinx cosh minus sinx over h plus the limit h goes to 0 cos x sin h over h okay so the left one and the right one 30:34 now i can factor a sinx out of that sinxtimecosh-1/h plus limit as h goes to 0 cosx times sinh 31:00 over h well the x terms dont contain h's they just stay the way they are, when x goes to 0 the limit of cosh-1/h is this limit which is 0 this part becomes sinx times 0 and here again cosx nothing happens when it goes to 0 31:30 the limit as h goes to 0, sinh/h is 1 what have we learned we learned that the derivative of sinx is cos so if y is sinx dy/dx cosx, you guys can remember that without to much work either 32:10 derivative of sinx is cosx almost so now i wont prove this one but the derivative of cosx is the negative of the sin of x 32:49 this top graph is the graph of sinx which you all memorized long time ago 33:00 lets think about what that looks like well whats the slope here remember now x goes to 0, sinx over x is 1 x and sinx are kind of doing the same thing and therefore the slope here is about 1 and now the slope is 0 about hre 33:30 so we have positive values until we get to 0 now lets look at the slope, theyre negative numbers until you get to here where its back to 0 and then theyre positive numbers again and then repeat thats the cos graph so the derivative of sin is the cos isnt that awesome, you guys are like math people thats really goof 34:10 these are important to memorize alright how bout some more fun ones lets do the derivative of tanx 34:35 how can we do the derivative of tanx, im not gonna do it the hard way ti much work, besides i did all that work to do the derivative of sin well tanx is sinx over cosx so if i wanted to do the derivative i use the quotient rule 35:02 lidehi/hidelo over lo squared derivative of the bottom cosx times the derivative of the top well we just learned whats the derivative of sin, cos minus the top sinx times the derivative of cos which is -sinx 35:30 all over bottom squared cos sqaured x this is cos squared minus, minus plus sin squared everyone remember what sin squared plus cos squared equals 1 thats a cos squared, thats just my handwritng 36:11 thats 1/cos squared also known as secant squared if y is tangent of x the derivative is secant squared how are we doing memorizing so far, the derivative of sin is cos 36:33 the derivative of cos is negative sin the derivative of tangent is secant squared 37:05 lets do the derivative of cotanx now we write it as cos/sin alright take a minute lets see if you can figure this out dy/dx is lo 37:30 dihi, derivative of cos is minus sin minus hidelo cosx cosx over sin squared x that is negative sin squared minus cos squared over sin squared 38:02 factor out the minus 1 and you get minus sin squared plus cos sqaured over sin squared which is negative 1 over sin squared which is negative negative cosecant squared 38:53 two left suppose we have y=cscx 39:01 well that 1 over cos this is minus sin squared minus cos squared take out a negative you get sin squared/ cos squared thats negative 1 over sin squared negative csc squared how do we do the derivative of tht one well we havent learned other stuff yet so lets do the quotient rule 39:40 lo derivative of 1 is just 0 minus hi time the derivative of the bottom over the bottom squared 40:09 cos x times 0 is 0 so this is just sinx over cos squared x but we can make that 1/cos times sin over cos we can think of it that way 40:32 1/c=csc sin/cos=tan we get y=secx dy/dx is secx tanx how are we going to remember that 41:19 1 more y=cscx so we can make this one 1/sinx 41:30 and then you get dy/dx equals sinx derivative of 1 which is 0 minus the derivative of sin 1 times cos over sin squared or negative cosx over sin squared x 42:02 you can now break that into negative 1/sinx times cosx/sinx or cscx cotx so last one y=cscx dy/dx is negative csc 42:31 cot so notice some patterns you take the derivative of the trig function if you take the derivative of its co function you get negtaive with the co functions when you have the derviative of sin the co sin co function the derivative is -sin which is the derivative of cos when you have tan you get csc when you get tan you get -csc sqaured 43:02 you get sec you get secx tanx when you get csc you get negative cscx cotx memorize these ill put some pages from my book up on blackboard soon alright lets do a little bit of practice nothing to hard, we save the hard stuff for the exams 43:42 whats the derivative of that lets do the derivative of e x sinx 44:02 well lets see the product rule first function times the derivative of the second function, whats the derivative of sin cos the derivative of e to the x is e to the x sinx that one wasnt so bad was ir 44:44 how bout that one what is the tanx over suqare root of x alright you ready, the derivative 45:00 low dehi, derivative of tan csc squared minus tangent times the derivative of the square root of x, did you memorize the short cut to that i hope you did, 1/2x over the bottom squared which is x lets add that to the pill of things we have to remember 45:37 i wouldnt let you take in a card with all this but feel free to tattoo the other one you have to memorize is this y=1/x dy/dx is negative 1/x squared 46:26 this is a fun one, lets take the derivative of this 46:58 this isnt very hard this is just regular stuff 47:02 so dy/dx is 20x cubed minus 6x squared the derivative if cot is csc squared so minus 2csc squared 6x thank you the derivative of cot is csc squared 47:30 the derivative of -cscx is the derivative of csccot so plus 4 csc cot remember when you take the derivative of csc and sac it returns to the derivative wait till we have to go backwards and do it the other way much harder that wont happen till 126, so that wont happen for a while alright i think thats enough for 1 day