Once again, another few problems I got stuck on in my practice set. Here is my work typed in Microsoft Word.
1: just negative sine no 4 simplifies to -4tanx
2: 2x^5 is what you derive for the second part not 2x^10
3: The derivative I am getting with CAS is 3*(sin5x)^{3x-1}*(sin5x)*ln(sin5x)+5xcos(5x) but I can't obtain that by hand.
Fullbox, . Don't treat it as if the power
was on theta, rather than being on outside. For clarity/simplicity, write it this way:
.
It might have been easier to note that, from the start, since by the properties of the
logarithmic function and , we have:
Oh okay thanks guys. I'm just horrible with syntax, that part really had me confused. I know you guys solved it already but I just want to update it onto the original work for the record's sake.
In regards to the third problem, I am still somewhat stuck. dwsmith, can you tell me what method you used in your post as opposed to typing out the answer? I want to try and figure it out myself I just don't know what to do. I'm unsure of what you did and in what order.
Yes (or at least I think so). I actually understood your first explanation a lot better than the second one with the logarithms.
From what I understood, you first find the derivative of the entire function and then you multiplied it by the derivative of the inner-most part of the function. My mistake was that I was multiplying the derivative of the entire function by the derivative of the inner-most function to the power of 4. In reality, I was not actually computing the derivative of the innermost function since I was misunderstanding the syntax (the exponent being before or after the variable for a trigonometric function, like sin^(2)(x) or sin(x)^2.)
I'm not so sure that I understand Soroban's way completely.
I understand the first step with the natural logs.
The next step I think I understand, but I'm not sure. I haven't reached the point where I can write down what I got without saying what I did in between so maybe that is what is confusing me.
From what it seems, we know ln(x) = 1/x. In this case, x=sin5x so he got 1/(sin5x). cos5x is the derivative of sin5x. 5 is the derivative of 5x. However, I am not so sure about what happens in the end of that line, where he Soroban writes 3*ln(sin5x).