How do you get this answer? (Differentiation)

[MonkeySage]

I am so confused right now.

f(x)=sqrt(x)
g(x)=x3+1

h(x)= (sqrt.(x))/(x3+1)

h'(x)= [(x3+1)*(1/2)x-1/2-sqrt(x)*3x2 ] / (x3+1)2

= [(1/2)x5/2+(1/2)x-1/2-3x5/2 ] / (x3+1)2
Where did the intact g(x) term go in the numerator?

= [(1/2)x5/2 + (1/2x) - 3x5/2 ] / (x3+1)2
Ok, I think I understand why the middle term changed.. sort of.

= [(1/2)x-1/2(x3+1-6x3 ] / (x3+1)2
What? How? Why? How did he go from the last step to this one?

= (1-5x3) / (2sqrt(x)(x3+1)2)
Everyone in class ended up with this answer, SOMEHOW, except for me. I have no idea what I'm missing, this makes absolutely no sense to me. I was pretty much lost after that g(x) term disappeared in step 2.


I am looking at this and wondering how on earth everyone in my class managed to get the right answer except for me; I asked the teacher but he wasn't much help at all at explaining how one manages to get (1-5x3) / (2sqrt(x)(x3+1)2) out of that...

Our instructor has this bad habit of assuming we're taking all the same mental steps that he is and doesn't write down all the intermediary steps, walking us through process and explaining why. He just jumps ahead with very little explaination.

[Grinner]

= [(1/2)x5/2+(1/2)x-1/2-3x5/2 ] / (x3+1)2
Where did the intact g(x) term go in the numerator?

Here, he distributed the (1/2)x-1/2 term among that g(x) term.

My work looks something like:
= [ (x3 + 1) * (1/2) x-1/2 - x1/2 * 3x2 ] / (x3 + 1)


= [(1/2)x5/2 + (1/2x) - 3x5/2 ] / (x3+1)2
Ok, I think I understand why the middle term changed.. sort of.

This isn't right. That middle term in the numerator should contain a square root. Clerical error?

From this step, you could combine the like terms in the numerator, multiply by one in the form of (2x1/2)/(2x1/2), and finish with some cancellation.


= [(1/2)x-1/2(x3+1-6x3 ] / (x3+1)2
What? How? Why? How did he go from the last step to this one?

Short answer: he multiplied the fraction by (2x1/2)/(2x1/2).

It looks like he did the reverse order of what I suggested above.

Here he first multiplies by one in the form of (2x1/2)/(2x1/2).
Then, he shifts the denominator's (2x1/2) up into the numerator, making the appropriate adjustments to its exponent's sign.
Last, he combines like terms in the numerator and shifts the (2x1/2) back down to the denominator. No, I can't see any reason why he moved it to the numerator in the first place.

[gomipile]

= [(1/2)x5/2 + (1/2x) - 3x5/2 ] / (x3+1)2
Ok, I think I understand why the middle term changed.. sort of.


The middle term shouldn't have changed.

What you need to do is collect like terms and factor out the simpler term in the numerator. That factor's reciprocal( 2sqrt(x) ) is nicer looking, so then you multiply numerator and denominator by that reciprocal at the same time.

[Broken Crown]

I am so confused right now.

f(x)=sqrt(x)
g(x)=x3+1

h(x)= (sqrt.(x))/(x3+1)

h'(x)= [(x3+1)*(1/2)x-1/2-sqrt(x)*3x2 ] / (x3+1)2

= [(1/2)x5/2+(1/2)x-1/2-3x5/2 ] / (x3+1)2
Where did the intact g(x) term go in the numerator?

It got multiplied by (1/2)x-1/2:

(x3+1)*(1/2)x-1/2

= x3*(1/2)x-1/2 +1*(1/2)x-1/2

= (1/2)x-1/2+3 + (1/2)x-1/2

= (1/2)x-1/2+6/2 + (1/2)x-1/2

= (1/2)x5/2 + (1/2)x-1/2


= [(1/2)x5/2 + (1/2x) - 3x5/2 ] / (x3+1)2
Ok, I think I understand why the middle term changed.. sort of.

That middle term should not have changed. It's a typo. Note that it changes back next line. This may be causing some of your subsequent confusion.


= [(1/2)x-1/2(x3+1-6x3 ] / (x3+1)2
What? How? Why? How did he go from the last step to this one?

He's removed the common factor (1/2)x-1/2 from the numerator, as follows:

(1/2)x5/2 + (1/2)x-1/2 - 3x5/2

= (1/2)x-1/2*((1/2)x5/2 + (1/2)x-1/2 - 3x5/2) / ((1/2)x-1/2)

= (1/2)x-1/2*((1/2)x5/2 + (1/2)x-1/2 - 36x5/2) / ((1/2)x-1/2)

= (1/2)x-1/2*(x5/2 + x-1/2 - 6x5/2) / (x-1/2)

= (1/2)x-1/2*(x5/2 + x-1/2 - 6x5/2)*(x1/2)

= (1/2)x-1/2*(x5/2+1/2 + x-1/2+1/2 - 6x5/2+1/2)

= (1/2)x-1/2*(x6/2 + x0 - 6x6/2)

= (1/2)x-1/2*(x3 + 1 - 6x3)

Which gives:

= (1/2)x-1/2*(1 - 5x3)

as the next step.

Putting the denominator back in, we move on to:

[(1/2)x-1/2(1-5x3 ] / (x3+1)2

= [(1-5x3*(1/2)x-1/2 ] / (x3+1)2

We can change the sign of the exponent of the bolded part and put it in the denominator:

= (1-5x3) / [2x-1/2*(x3+1)2]


= (1-5x3) / (2sqrt(x)(x3+1)2)
Everyone in class ended up with this answer, SOMEHOW, except for me. I have no idea what I'm missing, this makes absolutely no sense to me. I was pretty much lost after that g(x) term disappeared in step 2.

I hope that cleared it up a bit.


I am looking at this and wondering how on earth everyone in my class managed to get the right answer except for me; I asked the teacher but he wasn't much help at all at explaining how one manages to get (1-5x3) / (2sqrt(x)(x3+1)2) out of that...

Our instructor has this bad habit of assuming we're taking all the same mental steps that he is and doesn't write down all the intermediary steps, walking us through process and explaining why. He just jumps ahead with very little explaination.

Try being more assertive with your questions. I'm sure that you're not the only one in your class who would appreciate a little more explanation of the proofs, especially if your instructor is skipping the obvious steps. He's also probably taught this material a hundred times, so it seems routine and boring to him; he'll probably appreciate you showing an interest.

[MonkeySage]

I'm going to take another look at that problem... in the mean time, I went ahead with another one, linked here: http://tinyurl.com/pr93jh5

I used this website to check my answer, and apparently I'm completely wrong, which I don't get:

The original problem was g(x)=[(3x2+1)/(x+3)]2

I did this:
g'(x)=2[(3x2+1)/(x+3)] * [(x+3)(6x)-(3x2-1)2]/(x+3)2

= "" * (6x2+18x-6x2-2)/(x+3)2

= [2(3x2+1)*(18x-2)]/(x+3)3

= [(6x2+2)(18x-2)]/(x+3)3

[Grinner]

g'(x)=2[(3x2+1)/(x+3)] * [(x+3)(6x)-(3x2-1)2]/(x+3)2

You differentiated (x+3) incorrectly in the numerator. Highlighted for convenience.

[MonkeySage]

Well... I did get a different answer when I derived (x+3) correctly, but....

(Are images allowed?)
http://i1016.photobucket.com/albums/af282/anthonysherer90/20150923_2321351_zpsrei1izyl.jpg (http://s1016.photobucket.com/user/anthonysherer90/media/20150923_2321351_zpsrei1izyl.jpg.html)

[Grinner]

Well... I did get a different answer when I derived (x+3) correctly, but....

You silently dropped the parentheses in step 2 and consequently never distributed the negative 1.

You had the right idea in the rest of the steps, though.