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.
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.
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.
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.
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.