Indices, Exponential and Logarithmic Equations + Trig + Derivatives

[rushyrulz]

Re: Indices, Exponential and Logarithmic Equations + Trig Question

np broheim

[sickufully]

Re: Indices, Exponential and Logarithms Equations + Trig + Derivatives

I may or may not have one more question. But it's on derivatives. I'll assume you guys also know this stuff since you seem to know everything else.

We have to find dy/dx for each of the following.



Rules
if y = sinkx, then dy/dx = kcoskx
if y = coskx, then dy/dx = -ksinkx

The first one (vi) - I think it goes something like this:

y = 6sin(x/3) + 3cos(2x/3)
dy/dx = 6*(1/3)cos(x/3) + 3*(-2x/3)sin(2x/3)
dy/dx = 3/1cos(x/3) - 6x/3sin(2x/3)

Can I then simplify further to get:
dy/dx = 3cos(x/3) - 3x/1sin(2x/3)
or do I need to keep k as a fraction?



Rules
if y = e^kx, then dy/dx = ke^kx
if y = x^n, then dy/dx = nx^n-1

The second one (vii) - I didn't get very far. I think the first step is as follows, but I'm unsure of what to do next.

y = 3/x + e^3x/3 - 1/x^2
dy/dx = 3/1 + 3e^3x/3 - 1/2x
dy/dx = ???

[rushyrulz]

Re: Indices, Exponential and Logarithmic Equations + Trig + Derivatives

The first one (vi) - I think it goes something like this:

y = 6sin(x/3) + 3cos(2x/3)
dy/dx = 6*(1/3)cos(x/3) + 3*(-2x/3)sin(2x/3)
dy/dx = 3/1cos(x/3) - 6x/3sin(2x/3)

Can I then simplify further to get:
dy/dx = 3cos(x/3) - 3x/1sin(2x/3)
or do I need to keep k as a fraction?

You're on the right track here. The first half is totally correct, however it looks like you forgot to differentiate 2x/3 for the cosine part, that's the only problem. Oh, and 6/3 = 2, not 3

Rules
if y = e^kx, then dy/dx = ke^kx
if y = x^n, then dy/dx = nx^n-1

The second one (vii) - I didn't get very far. I think the first step is as follows, but I'm unsure of what to do next.

y = 3/x + e^3x/3 - 1/x^2
dy/dx = 3/1 + 3e^3x/3 - 1/2x
dy/dx = ???

3/x does not differentiate to 3/1, it would if x were in the numerator. If you leave x in the denominator, you will need to use the quotient rule which is quite messy, which is why we move them to the numerator by using negative exponents. If you do not know this rule, I suggest you brush up on it! You've differentiated the middle term perfectly.

[sickufully]

Re: Indices, Exponential and Logarithmic Equations + Trig + Derivatives

So with the first one, when you do this step:


What exactly happens to the 6 out the front and the +3 before (-sin...)?
Are they just divided by each other, so the 6 = 2 and the 3 = 1 (which just disappears)?

And for the second one, I completely missed that rule for 1/x = x^-1. Also, the final simplifying step you did, is it necessary to do that or is it OK to leave it as: -3/x^2 + e^3x + 2/x^3?

I do understand how you got 2-3x/x^3 + e^3x, I'm just wondering if I'll lose marks in a test for not simplifying further.

[rushyrulz]

Re: Indices, Exponential and Logarithmic Equations + Trig + Derivatives

So with the first one, when you do this step:


What exactly happens to the 6 out the front and the +3 before (-sin...)?
Are they just divided by each other, so the 6 = 2 and the 3 = 1 (which just disappears)?

6 * 1/3 = 2, 3 * 2/3 = 2. Distributive property moves the 2 outside the parenthesis.

And for the second one, I completely missed that rule for 1/x = x^-1. Also, the final simplifying step you did, is it necessary to do that or is it OK to leave it as: -3/x^2 + e^3x + 2/x^3?

I do understand how you got 2-3x/x^3 + e^3x, I'm just wondering if I'll lose marks in a test for not simplifying further.

You won't lose points on a test if you don't simplify it all the way down like that, but you'll look like one smart cookie if you do~

[Daniel Cudmore]

Re: Indices, Exponential and Logarithmic Equations + Trig + Derivatives

Yeah so this trimester I'm taking Math Methods (not by choice obviously) and since I pretty much failed what we call 'Further Maths' (the level below Methods) in yr12, I really don't see myself passing this subject unless I receive some assistance. There are some questions which have really confused me and aren't covered in the text book so if anyone knows how to solve these, please share your wisdom.

Ok so I can't believe I actually know what you are referring to - my cousin and I were just chatting on math methods. He was getting math methods help that worked for him - not sure if it will work for you - your miles may vary.

We were mostly debating as to what's the practical use for this course as he was thinking about an engineering degree, and one of his class mate want to be an art history major, lol.

[Travis_Flesher]

indices, exponential and Logarithmic Equations, trigonometry and Derivatives

Quote ~ sickufully
I really don't see myself passing this subject unless I receive some assistance.
If you want to pass you've just got to play the trump card man.
if anyone knows how to solve these, please share your wisdom.
As a former algebra student I don't see where enough information is provided in this post to actually solve these variable formulas but they are certainly very interesting nonetheless, regardless of what they actually mean.
The following questions are the ones I'm having trouble with. I've attempted them but my answers just don't look right.
I agree that they don't look right, but what's important is thinking critically, being creative, and communication. My advice, and I'm just guessing here, but just show as much work as possible.

To sum it up, it looks to me like the goal here is to either attempt to simplify old formulas just to get the ol' brain firing, or optimally, to actually simplify an old formula.
To succeed in simplifying a preexisting physics formula is like gold to a surprisingly large demographic of math students, but the motherlode is in figuring out engineering/physics formulas from scratch for whatever reason usually to streamline some aspect of application or for reference. These practical formulas, such as force produced from different types of fuel combustion, how fast a hot air balloon flys away as more heat is pumped in, or the rate at which rubber tires degrade as driven over asphalt for instance often end up becoming fodder for a growing number of college kids around the world. Sorry I probably didn't help you much sicku, but I'm glad you posted this I thought it was cool as heck! Thank you buddy.