Question

Find all the values of x for which the function is differentiable.

h(x)=cuberoot(3x-6)+5

Answer 1

\[\sqrt[3]{(3x-6)}+5\]

Answer 2

I think it's a continuous function itself ....it's a cube root unlike square roots cube can have a minus sign too. So all Real Numbers

Answer 3

at x = 2 the equation has many tangents.

Answer 4

@Chaise, do you know that after graphing it?

Answer 5

Well - I graphed it and then looked for a relationship.

3x-6; x = 2
3x2 -6
6-6 = 0
The equation has no tangents when the cube root sign has a 0 under it? Does this make sense?

Answer 6

@Chaise, do you know that after graphing it?

Answer 7

Where did you get x = 2?

Answer 8

Disregard the +5 in the equation - this is just changing the y intercept and has no relevance to the derivative as the derivative of a constant becomes 0 anyway.

Answer 9

@Chaise, do you know that after graphing it?

Answer 10

Are you saying when x equals 2? So, you are looking at a graph?

Answer 11

Yes - when x = 2 then the terms under the cubed root become 0. I assume you cannot take the derivative when there is a 0 in radical.
http://www.wolframalpha.com/input/?i=h%28x%29%3Dcuberoot%283x-6%29

Answer 12

And this makes sense, logically, at least in my opinion.

Answer 13

@Chaise, do you know that after graphing it?

Answer 14

Oh I see. Just for review, \[\sqrt[3]{3x-6}\].. how is it "moved" by looking at the numbers in the function? I forgot...

Answer 15

For example, I know what \[\sqrt[3]{x}\]looks like, so how would I "move" it to match the one in the problem?

Answer 16

I don't really know what you're asking.
\[\sqrt[3]{x}=x ^{(1/3)}\]
\[\sqrt[3]{=3x-6} = (3x-6)^{1/3}\]
Now you can differentiate it using the power and chain rule. Need help with this?

Answer 17

Disregard the = sign in the cube root there, sorry.

Answer 18

I mean like, for example, if I memorize the graph of x^2, I would know how x^2+4 looks like... I just graph x^2 and shift it 4 units up... that's what I am asking.

I know what the graph of \[\sqrt[3]{x}\] looks like.. how do I figure out what \[\sqrt[3]{3x-6}\] looks like using the translations.

Answer 19

So when you take the derivative you get:
(3(x-2))^(1/3)
When you make x equal to 2, you get 0 (I did this on my calculator). Thus the equation is not differentiable at x = 2

Answer 20

How do you graph it? Let a computer graph it, or substitute values in for x and then plot them on a cartesian plane.

Answer 21

Whenever the derivative equals 0, it means it is not differentiable?

Answer 22

I thought the derivative is the slope, so even if it equals 0, it means that the tangent is a horizontal line?

Answer 23

yes - that's right. however, look at the graph on this website (that's your function) and I assure you, one of the tangents is horizontal, but there are many more which can be drawn.
http://www.wolframalpha.com/input/?i=h%28x%29%3Dcuberoot%283x-6%29%2B5

Answer 24

Notice how there is multiple tangents?

Answer 25

Yes, ohh I see. My teacher will not allow us to use graphing calculators on the test, which is why I don't want to be too dependent on graphing utilities. In the case of a test, my safest bet would be to plug in points and graph it?

Answer 26

I'm not too sure - best off ask your teacher on this one. Afterall - she marks your test. I'm allowed full access to a calculator (pretty much wolfram alpha) in all my tests.

Answer 27

Okay, thank you for the help.