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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

[00:05:03] just start with a power rule
problem

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?

[00:51:55] sorry about this

[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