Differential calculus help?

Factual Questions
1

Just getting into differential calculus for the first time. I’m working with the Product Rule and Quotient Rule right now. I’m not just throwing homework at you, I really need input on these concepts that my teacher fails to clarify.

I’m getting stuck repeatedly on these problems, so please help and point out any mistakes I’m not seeing. Here’s an example I’ve been working over, because it includes a square root:

h(s) = √s(4-s^2)

Now I do this:

s^1/2(4-s^2)

s^1/2(2s) + (1/2s^-1/2)(4-s^2)

And I’m clueless as to what to do at this point. I’m told that to simplify, I must find the common factor with the smallest valued exponent, but I see no common factor. Does this need to be simplified?

I’m sorry if I format math terribly. Any help would be greatly appreciated.

2

[del]Are you trying to use this:[/del]

Nevermind, that didn’t make any sense.

I’m having a hard time following your notation.

3

Your notation is fine to me, though you’ll want to parenthesize or superscript some of those exponent "1/2"s for clarity, I suppose.

Anyway, take -s^(1/2)(2s) + (1/2s^-1/2)(4-s^2) [I’ve reinserted a negative sign on the left term, which you accidentally forgot; this is because the derivative of 4 - s^2 is not 2s, but rather -2s]. What was I saying? Right, take -s^(1/2)(2s) + (1/2s^(-1/2))(4-s^2) and expand out the multiplication on the right, as well as remember that when you multiply two different powers of s, you can add the exponents. So you get -2s^(3/2) + 2s^(-1/2) - (1/2)s^(3/2). Now you can combine the two different terms with the same exponent, to get -2.5s^(3/2) + 2s^(-1/2). And that, presumably, is good enough to stop with.

Your actual calculus (the differentiation using product rule) was perfectly fine [except for missing one negative sign]. No need to worry about any of that. It’s just the algebraic mungling around at the end (“simplification”, though of course what counts as simplified is semi-subjective) that you didn’t finish.

4

Well if I said that example out loud, it would be “The square root of s; parenthesis, 4 minus s squared, end parenthesis.”

I then convert the square root into an exponent of 1/2. I find the derivative according to the Product Rule:

h’(x) = (f(x))(g’(x)) + (f’(x))(g(x))

Am I misusing the rule or is there another error? I get the same problem with any other example I create to try out.

5

You used the rule fine [except for missing one negative sign]. As I said above, you haven’t made any big mistakes; you just didn’t go all the way in “simplifying” at the end. What you were missing, I suppose, is that when you multiply two different powers of the same term, the result is that term raised to the sum of those powers. [E.g., x^3 * x^5 = x^8, and, in your case, s^(-1/2) * s^2 = s^(3/2)]. Once you take that into account, you can move things around a bit, into something which might be considered simpler.

6

Oh, okay, you actually multiply using FOIL in simplification? A familiar face!

I think I understand the process well enough now. Thank you, Indistinguishable.

7

Although, say in this section:

-s^(1/2)(2s) + (1/2s^(-1/2))(4-s^2)

You fit in trigonometric values:

-s^(1/2)(cos x) + (1/2s^(-1/2))(sin x)

How would the multiplication expand? FOIL can’t apply, can it?

Or is it already “simplified” completely? As you can see, my math is poor.

8

Yes, the second one is simplified pretty much the most it could be.

The good news is, your understanding of calculus looks quite good. Your difficulties only seem to be with some of the algebra - just remember it isn’t any different than what you’ve probably seen before. I wouldn’t focus on it too much. Although it is quite useful to know how to manipulate expressions into various forms, it hopefully isn’t something you’re going to get marked down for.

9

I’d just like to note that I had the exact same struggle when I first took calculus in high school. I understood how to apply the differentiation and integration rules just fine, it was always the algebra (especially factoring, which I still can’t do correctly) that messed me up.

Calculus is a lot easier than algebra, once you get it.

Hang in there.

10

I don’t see anywhere you’re multiplying a binomial by another binomial, which is when FOIL would apply.

Here’s what you have after applying the Product Rule:

–s[sup]1/2/sup + (1/2s[sup]-1/2[/sup])(4-s[sup]2[/sup])

Using the Distributive Property, and the fact that a[sup]m[/sup]a[sup]n[/sup] = a[sup]m+n[/sup], you get

–2s[sup]3/2[/sup] + 2s[sup]-1/2[/sup] - (1/2)s[sup]3/2[/sup]

You can combine the s[sup]3/2[/sup] terms, and, if desired, factor out s[sup]-1/2[/sup].

An alternative approach that avoids the Product Rule is to multiply things out before differentiating:

h(s) = 4s[sup]1/2[/sup] - s[sup]5/2[/sup]