# Tutor.com The Derivative Session

##### Oct. 16, 2012

Session Transcript - Math - Calculus, 10/11/2012 11:09PM - Tutor.comSession Date: 10/11/2012 11:09PM
Length: 71.5 minute(s)
Subject: Math - Calculus

System Message
[00:00:00] *** Please note: All sessions are recorded for quality control. ***

Guest (Customer)
[00:00:00] Hi! Can you give me derivative problems to practice that we can do? Thanks!

Jason P (Tutor)
[00:00:18] Hello! Welcome.

Guest (Customer)
[00:00:45] Involving Tangent lines, power rule, product and quotient rule, and chain rule
[00:00:48] Hello!
[00:00:50] thank you.

Jason P (Tutor)
[00:01:21] Sure, let's see what I can think up...

Guest (Customer)
[00:01:41] thanks!

Jason P (Tutor)
[00:04:54] Ok. So let's try to find where the horizontal tangent lines are for the function f(x) = x 3 sin x

Guest (Customer)
[00:04:57] oh we haven't exaclty done that

Jason P (Tutor)
[00:05:20] ok, no problem
[00:06:21] let's find the derivative of 3 x 3 - 10 x

Guest (Customer)
[00:06:11] ok
[00:06:17] is that x^3
[00:06:17] ?

Jason P (Tutor)
[00:06:35] yep

Guest (Customer)
[00:06:33] 9x^2-10

Jason P (Tutor)
[00:06:56] huzzah!
[00:07:12] so power rule doesn't really have any tricks up it's sleave

Guest (Customer)
[00:07:03] right
[00:07:12] can we do one with a fraction
[00:07:17] where I have to re-write
[00:07:21] or square root

Jason P (Tutor)
[00:07:35] Let's look at quotient rule...yep

Guest (Customer)
[00:07:25] or one of each
[00:07:32] with power

Jason P (Tutor)
[00:09:13] Let's try finding the derivative of squarroot[(2 x )^3]

Guest (Customer)
[00:09:15] can you write on whiteboard
[00:09:22] with square root
[00:11:10] ?

Jason P (Tutor)
[00:11:35] That's the best way, yes

Guest (Customer)
[00:11:34] and now do I do the chain rule
[00:11:34] or
[00:11:38] can we do it togeether

Jason P (Tutor)
[00:12:07] you could - but it'd be easier to multiply the exponents
[00:12:17] then use power rule once

Guest (Customer)
[00:12:14] oh ok

Jason P (Tutor)
[00:13:38] woops! sorry to write over you!

Guest (Customer)
[00:13:34] how can you do power since it's in parentheses
[00:13:38] wouldn't you do chain

Jason P (Tutor)
[00:14:12] we could, but note another property of exponents....

Guest (Customer)
[00:14:15] right
[00:14:15] but

Jason P (Tutor)
[00:14:50] so we don't need to use power rule here

Guest (Customer)
[00:14:41] ok

Jason P (Tutor)
[00:14:57] we could, but it isn't necessary

Guest (Customer)
[00:14:47] oh
[00:14:51] so just write like that
[00:14:56] so that's the derivative

Jason P (Tutor)
[00:15:20] nope! we still need to use the power rule

Guest (Customer)
[00:15:21] okay just kind confusing
[00:15:34] ok what next

Jason P (Tutor)
[00:16:20] now we bring down the exponent to be a coefficient, and reduce the power by one:

Guest (Customer)
[00:16:43] oh ok
[00:17:09] wait
[00:17:11] what did you do
[00:17:20] how'd you get x^1/2

Jason P (Tutor)
[00:19:03] see how I did that?

Guest (Customer)
[00:19:00] this is just a bad problem...
[00:19:17] because chain rule would be a better choice here

Jason P (Tutor)
[00:19:50] you want to try it that way?

Guest (Customer)
[00:20:20] what way
[00:20:24] oh
[00:20:25] yeah
[00:20:26] sure
[00:21:21] is that right?

Jason P (Tutor)
[00:21:53] it is equivalent! we have success

Guest (Customer)
[00:21:48] yay
[00:21:48] ok

Jason P (Tutor)
[00:22:05] let's try a quotient rule?

Guest (Customer)
[00:21:54] this is so much easier
[00:22:03] sure

Jason P (Tutor)
[00:22:19] yeah, I love chain rule

Guest (Customer)
[00:22:24]
[00:22:25] it's fun
[00:22:29] can we do another chain
[00:22:38] harder one involving trig functions

Jason P (Tutor)
[00:22:53] sure
[00:24:14] enough? too much?

Guest (Customer)
[00:24:15] eh yeah too much
[00:24:27] well I can try...

Jason P (Tutor)
[00:24:49] whichever

Guest (Customer)
[00:24:41] ok we can do this first
[00:25:18] so far so good?

Jason P (Tutor)
[00:26:04] sure

Guest (Customer)
[00:26:12] ?
[00:26:29] I forgot what to do now

Jason P (Tutor)
[00:27:06] ok, so the most outer function is the fifth power...

Guest (Customer)
[00:27:37] is that something like it?
[00:27:44] idk

Jason P (Tutor)
[00:28:28] we first bring the 5 down, but the sine stays...

Guest (Customer)
[00:28:29] oh why

Jason P (Tutor)
[00:30:03] well, with chain rule, we need to apply the derivative to each nested function one after another

Guest (Customer)
[00:30:22] oh right
[00:30:23] ok
[00:30:50] oh okay
[00:30:55] can we do another like this

Jason P (Tutor)
[00:31:15] sure
[00:32:46] how's that?

Guest (Customer)
[00:34:29] ok
[00:34:30] ?

Jason P (Tutor)
[00:35:11] so long as your 1/2 exponent is negative and your sine is negative as well
[00:35:25] Looks good!

Guest (Customer)
[00:35:14] wait
[00:35:15] why
[00:35:25] 1/2 in the front should be neg?
[00:35:56] oh
[00:35:56] nvm
[00:36:01] got what you're saying
[00:36:03] sorry abotu that

Jason P (Tutor)
[00:36:22] the whole expression is negative because the derivative of cos x is....no problem

Guest (Customer)
[00:36:12] can we do a couple more of quotient and product please

Jason P (Tutor)
[00:36:36] sure

Guest (Customer)
[00:37:21] is that x^3 at the bottom?

Jason P (Tutor)
[00:37:40] yep

Guest (Customer)
[00:39:57] is that right?

Jason P (Tutor)
[00:40:29] sure is!
[00:41:08] do you have any questions about doing different derivatives?
[00:41:20] what we've done so far or still could do?

Guest (Customer)
[00:41:12] are all the signs right

Jason P (Tutor)
[00:41:38] oh....

Guest (Customer)
[00:41:33] I just have trouble with the trig derivatives involving chain rule

Jason P (Tutor)
[00:43:28] how 'bout that one?

Guest (Customer)
[00:43:20] for the previous one...

Jason P (Tutor)
[00:43:40] yes?

Guest (Customer)
[00:43:27] should it be
[00:43:31] +1
[00:43:39] 2x-1 or +1?

Jason P (Tutor)
[00:44:01] 2x - 1

Guest (Customer)
[00:43:54] oh okay
[00:44:13] wait how do you know that
[00:44:16] can you show me

Jason P (Tutor)
[00:44:43] sure

Guest (Customer)
[00:44:56] oh I see it
[00:44:56] okay

Jason P (Tutor)
[00:45:21] ok, good

Guest (Customer)
[00:45:14] for the next one
[00:45:23] do I use product or what?

Jason P (Tutor)
[00:45:44] chain

Guest (Customer)
[00:45:29] how can I use chain?

Jason P (Tutor)
[00:45:59] first find the derivative of sine

Guest (Customer)
[00:45:56] ok
[00:46:06] ohh
[00:46:51] wait is that right

Jason P (Tutor)
[00:47:22] yep!

Guest (Customer)
[00:47:34] oh okay
[00:47:35] gotcha
[00:47:37] can we do another

Jason P (Tutor)
[00:48:04] yes

Guest (Customer)
[00:49:16] so first we find the derivative of sin right

Jason P (Tutor)
[00:49:55] I would first take the derivative of the power..

Guest (Customer)
[00:50:24] ok
[00:50:33] is that right
[00:50:53] ohhh
[00:50:55] wooops
[00:51:21] oh
[00:51:39] can the derivative be taken for the second?
[00:51:41] sin

Jason P (Tutor)
[00:51:58] your second term is right

Guest (Customer)
[00:51:48] ok
[00:51:49] figured
[00:51:50] ok
[00:51:52] one more?
[00:52:00] just want to get it stuck in my head

Jason P (Tutor)
[00:52:37] no problem!

Guest (Customer)
[00:54:27] like can we do one involving chain rule

Jason P (Tutor)
[00:55:11] ok...let's see one with a little more

Guest (Customer)
[00:56:58] ok
[00:57:16] so first what to do
[00:57:17] um

Jason P (Tutor)
[00:57:45] the outermost function is tan
[00:57:56] so we take the derivative of tan first

Guest (Customer)
[00:57:46] oh
[00:57:48] so is this right
[00:57:49] or no

Jason P (Tutor)
[00:58:07] no
[00:59:04] good!

Guest (Customer)
[00:58:50] right so far?
[00:59:12] ugh..

Jason P (Tutor)
[00:59:36] so now for the inside...

Guest (Customer)
[00:59:39] um
[00:59:48] tka ethe derivative of cos correct?
[01:00:25] ?

Jason P (Tutor)
[01:00:45] inside the tan function there is a product...so let's do this...

Guest (Customer)
[01:00:38] oooh
[01:00:38] right
[01:00:39] ok
[01:02:07] so that's the answer right

Jason P (Tutor)
[01:02:26] that's it!

Guest (Customer)
[01:02:40] can we do one ast one
[01:02:41] last

Jason P (Tutor)
[01:02:56] so we always work our way from the outside in

Guest (Customer)
[01:03:06] right
[01:04:22] is that it?
[01:06:45] ok
[01:06:47] ??
[01:08:02] oh wow
[01:08:23] wait how'd you do that

Jason P (Tutor)
[01:08:53] let's color code this!

Guest (Customer)
[01:09:42] ooh right
[01:09:43] ahh

Jason P (Tutor)
[01:10:19] yep!

Guest (Customer)
[01:10:06] ahhh
[01:10:14] gotcha
[01:10:33] okay thank you!
[01:10:37] gunna go to bed now
[01:10:38] haha

Jason P (Tutor)
[01:10:55] you're welcome!
[01:11:04] good luck and keep practicing

Guest (Customer)
[01:10:53] bye
[01:10:58] !!

Jason P (Tutor)
[01:11:13] bye!

Guest (Customer)
[01:11:00] thanks