Basic simplifying question in derivatives (Find derivative of P(t) = [(t+3)^(1/2)]/[(t-3)^(3/2)])

[dsber]

I wasn't sure where to post this because it's sort of an algebraic problem I'm having . Here is the prompt



So I found that I'm struggled in these questions because I wasn't simplifying correctly.



I have no idea how the fifth line simplifies to the last line. Specificially I don't know how it gets to [(t-3)-3(t+3)]. If you have any resources you can share I would love to fill this gap in my knowledge.

 

[Dr.Peterson]

I wasn't sure where to post this because it's sort of an algebraic problem I'm having . Here is the prompt

View attachment 35465

So I found that I'm struggled in these questions because I wasn't simplifying correctly.

View attachment 35466

I have no idea how the fifth line simplifies to the last line. Specifically I don't know how it gets to [(t-3)-3(t+3)]. If you have any resources you can share I would love to fill this gap in my knowledge.

What they've done there is to factor out the factors they show in the numerator.

That is, what do you have to multiply

by to get
?

This may be a little different from factoring you have done before, but it is the same idea.

There is more to do after the last step you show, of course.

 

[Otis]

I have no idea how the fifth line simplifies to the last line.

Hi dsber. The fifth line shows a common denominator, so those two ratios may be combined. Next, it appears that they've factored out the common factors of (1/2), (t+3)^(-1/2) and (t–3)^(1/2).
[imath]\;[/imath]

 

[dsber]

Thanks for the responses. I think what I'm having trouble understanding is how they factored everying on the right side. For example where does the 1/2 exponents go, and where does the 3/2 go. They don't multiply everything by 2. I think I'm fundamentally missing some knowledge so apologies if this looks like dumb question.

 

[Steven G]

In the line where they have the subtraction of two fraction, they factored out (1/2)(t-3)^1/2(t+3)^-1/2.
Why should (t-3)^1/2 be factored out? Because in the 1st fraction you have a factor of (t-3)^3/2 and in the 2nd fraction you have a factor of (t-3)^1/2. As always you factor out the one with the smaller power and that is 1/2.

Can you go from there?

 

[Otis]

I think I'm fundamentally missing some knowledge

Hi. Here's a short example of using a property of exponents (and a substitution of symbols) to help factor.

Let's factor the expression (t+3)^-½ – (t+3)^½

Here's the property of exponents: u^m ∙ uⁿ = u^m+n

Therefore, when factoring a term like u^m+n, we know it takes the form u^m ∙ uⁿ, and if we have values for m and m+n, then we can find n for our factorization.

I'll make a substitution in the example expression to factor, for simplified reading.

Let u = (t+3)

So, we have u^-½ – u^½ and we want to factor out u^-½.

u^-½ (?? – ??)

What terms do we need inside the grouping symbols, so that expanding the factorization yields back what we're factoring? Well, it's clear that the first term must be 1.

u^-½ (1 – ??)

To get that second term, we can think in terms of the exponent property.

u^m ∙ uⁿ = u^m+n

u^-½ ∙ u^?? = u^½

From that, can you see that -½ plus ?? must equal ½? In other words, -½+n=½ so we need n=1.

The factorization is u^-½ (1 – u), and we have:

(t+3)^-½ – (t+3)^½ = (t+3)^-½ (1 – t – 3)
[imath]\;[/imath]

 

[Dr.Peterson]

Thanks for the responses. I think what I'm having trouble understanding is how they factored everying on the right side. For example where does the 1/2 exponents go, and where does the 3/2 go. They don't multiply everything by 2. I think I'm fundamentally missing some knowledge so apologies if this looks like dumb question.

You should see it more easily if you read that step in reverse, multiplying each of the [imath](t-3)[/imath] and the [imath]3(t+3)[/imath] by the numerator shown.

Factoring is a matter of working backward, so you need the experience of working forward (multiplying) to get used to what to expect in factoring.

 

[Steven G]

Thanks for the responses. I think what I'm having trouble understanding is how they factored everying on the right side. For example where does the 1/2 exponents go, and where does the 3/2 go. They don't multiply everything by 2. I think I'm fundamentally missing some knowledge so apologies if this looks like dumb question.

Think about factoring this:

x⁵- x² = x²(x³-1).
Where did 5 and 2 go? Whenever you factor out x raised to a power, the powers generally change.
Hint: It's time to recall how to add/subtract fractions.

 

[limiTS]

Maybe it'll help if the term being factored out is made explicit. Just the numerator:

[imath]\displaystyle (t-3)^{\color{red}3/2}\cdot\frac{1}{2}(t+3)^{-1/2}-(t+3)^{1/2}\cdot\frac{3}{2}(t-3)^{1/2}[/imath]

Well, [imath]\displaystyle{\color{red}\frac{3}{2}}=1+\frac{1}{2}[/imath]

[imath]\displaystyle (t-3)^{\color{red}1+\frac{1}{2}}\cdot\frac{1}{2}(t+3)^{-1/2}-(t+3)^{1/2}\cdot\frac{3}{2}(t-3)^{1/2}\\ \displaystyle (t-3)^{1}\cdot(t-3)^{\color{red}1/2}\cdot\frac{1}{2}(t+3)^{-1/2}-(t+3)^{\color{blue}1/2}\cdot\frac{3}{2}(t-3)^{\color{red}1/2}[/imath]

Now the [imath]\displaystyle (t-3)^{\color{red}1/2}[/imath] can be factored out.

As for the [imath](t+3)[/imath]'s, think of [imath]\displaystyle (t+3)^{\color{blue}1/2}[/imath] as [imath]\displaystyle (t+3)^{1\color{blue}-\frac{1}{2}}=(t+3)^1\cdot(t+3)^{\color{blue}-1/2}[/imath]

Then the numerator looks like:

[imath]\displaystyle (t-3)^{1}\cdot(t-3)^{\color{red}1/2}\cdot\frac{1}{2}(t+3)^{\color{blue}-1/2}-(t+3)^1\cdot(t+3)^{\color{blue}-1/2}\cdot\frac{3}{2}(t-3)^{\color{red}1/2}[/imath]

Factor out both the [imath](t-3)^{\color{red}1/2}[/imath] and the [imath](t+3)^{\color{blue}-1/2}[/imath] to get:

[imath]\displaystyle (t-3)^{\color{red}1/2}\,(t+3)^{\color{blue}-1/2}\left[(t-3)\cdot\frac{1}{2}-(t+3)\cdot\frac{3}{2}\right][/imath]

I'll let you figure out how to factor out the [imath]\displaystyle \frac{1}{2}[/imath]

 

[dsber]

Ohhh, I finally get. I've dealt with this kind of example before but just never really understood it when it came to exponents. Thank you so much everybody for clearing this up for me!