### Stony Brook MAT 125 Spring 2015 Lecture 13: Derivative of logarithmic functions March 23, 2015

 Start know how excited you are so how would I find the natural log of x so by the way when i say log thats what i mean the only log you really care about is natural log we rewrite this as 0:30 e to y because all of you who had me last semester know they are inverse functions now lets take the derivative, derivative x is 1, derivative of e to the y is e to the y dy/dx because of the chain rule and because its implicated divide by e to the y 1:03 and you have 1/e to the y but e to the y is just x you get 1/x so if y is natural log of x derivative is 1/x so this is where it comes from for those of you who care and this is the answer 1:31 part of the math class im suppose to show you were it comes from you can do that its not that hard now you wouldnt have log with messy stuff inside so a nice short cut if you have log of a function youd do the derivative write a fraction bar put the function on the bottom put the derivative on top 2:03 thats the chain rule version of that for example suppose i had y=ln of x squared plus 3x plus 1 and the derivative function goes on the bottom derivative goes on top so function on the bottom 2:32 derivative on top just that simple your waiting for it to get trickier, its not trickier lets do another one suppose i had y=ln cosx dy/dx is function on the bottom 3:00 derivative on top which by the way is negative tan of that funny how that happens you mean if i gave you this on an exam either of those are acceptable would be nice to know you can get from here to here but you dont have to 3:32 yes iw ould not take off for missing minus sometimes i do, depends on my mood so thats how you do the derivative of natural log what if its not natural log but its log of some other base heres something you probably dont remember 4:06 in 123 we taught you change in base rule which means you can write this log base a of x over log base a of b or in this case natural log of x over natural log of b a is some other base, thats called the change in base rule 4:31 so because you could write log of some base b of x lnx over natural log of b so natural log of b is a constant you can think log base e of b as 1/ln of b lnx remember log of b is just a constant if you want tof ind the derivative 5:00 of the log of any base other than e same derivative still 1/x and now you just have the natural log of b in ther, so now it just becomes this and the same with the chain rule so if you had y is log base b of a function of x and the derivative 5:34 is gonna go on top, function on the bottom natural log of the base for example you have y= common log, log base 10 of x squared minus 4x, the derivative 6:01 ts exactly what you would do if it was natural log x squared minus 4x on the bottom youd put 2x-4 on top then youd just have to multiply by natural log of 10 why 10? because thats log base 10 thats because its log base 10 6:40 thats 2 down, 5 to go how we doing so far? 7:13 we learned how to do a derivative e to the x what if we had something other than e to the x what if we had b to the x how would we find the derivative of that 7:32 well you could write this as e to the natural log of b to the x and you remember the log rules that is e to the x natural log of b, remember log b is just a number take the derivative 8:02 would be e to the x natural log of b times natural log of b and this is just b to the x so another rule i got room over here y is b to the x and the derivative b to the x log b so another words 8:30 with the logarithm youre going to divide by the natural log of b with the power youre going to multiply by the natural log of b in second semester calc you do it backwards to keep you on your toes suppose i wanted to do y=7 to the x then the derivative is 7 to the x, just like e to the x 9:00 times ln7 this isnt so bad, theres much worse stuff out there is i had y is 10 to the x squared plus 4x-5 the derivative 9:33 10 to the x squared plus 4x-5 thats the first part nothing changes times the derivative of x squared plus 4x-5, which is 2x plus 4 ln10 so just like you did e but except you have a natural log piece in the end 10:08 so if i had y=e to the sinx the derivative is e to the sinx times cosx if i had 10:30 y=5 to the sinx then the derivative is 5 to the sinx times cosx times ln5 how we doing so far? 11:09 how we doing so far this isnt so bad lets practice some more of these 11:53 alright heres a problem you should be able to do 16:34 alright lets take the derivative of this so you want to find the tangent line you do y-y1 is slope times x-x1 we know x1 is 3 so we need to find y1 how do we find y1, we do f of 3 17:01 is the ln 3 squared which is 9-8 thats log1 and log 1 is 0 so now we have y-0 is m times x-3 we just need to figure out what m is were in calc class the only thing you need to know to do is take the derivative so take the derivative the derivative is 17:32 x squared minus 8 goes on the bottom 2x goes on top evaluate this at 3 you get 6 over 9-1 which is 1 so you get 6 y is 6 times x minus 3 howd we do happy no no 18:03 when you plug that in that gives you the slope and the y coordinate you plug in the original function you get the y coordinate you plug in the derivative you get the slope lets do another one of these 19:19 okay same idea find the eqaution of a tangent line, we like these for exam questions we like theses equations of a tangent line eqautions 22:13 alright lets do this one 22:34 we need to do the y-y1= blah blah so y-y1 =m(x-2 you plug 2 in for y you get the y coordinate so y1 is going to be 6 is going to be 2 which is 36 23:00 where does the 2 come from 8-6 is 2 so you have y-36 m times x-2 now lets take the derivative the derivative is dy/dx is 6 to the the 4x-6 that doesnt change the derivative of 4x-6, which is 4 23:30 times the ln6 which is something like the natural log of 6 now we have to do at x=2 youre going to get 6 to the 2 times 4 times log 6 thats 144 log 6 24:04 so your equation is y-36 is 144 ln6 times x-2 got the idea? wednesday we are going to do more derivative stuff