Rearranging Euler's number in Chain Rule

[raveyp]

I don't understand how to get from the top step to the next. Am I missing some ruler with Euler's number?

When I try to work through it using order of operations I end up with -1/3e^m + 1/3e^2m

Wouldn't 1/3e^2m end up as 1/3(Cos(2)+mSin(2))?

 

[HallsofIvy]

Your attachment doesn't seem to have anything to do with the problem posed- but it is hard to tell what your attachment is supposed to mean since there are parentheses, "( )", without anything in them.

I guess you are trying to use [itex]e^x= cos(x)+ I sin(x)[/itex] although I don't see any "i" in what you have.
[itex]e^m= cos(m)+isin(m)[/itex] and [itex]e^{2m}= cos(2m)+ isin(2m)[/itex]. But then you have the "m" outside the trig functions. I don't see any way or doing that. Perhaps you could explain better what you are trying to do?

 

[raveyp]

The image is a clip from an intro to calculus for machine learning video. I'm just trying to understand how they've gotten the dh/dm by multiplying out the e^m.

 

[pka]

I don't understand how to get from the top step to the next. Am I missing some ruler with Euler's number?
When I try to work through it using order of operations I end up with -1/3e^m + 1/3e^2m
Wouldn't 1/3e^2m end up as 1/3(Cos(2)+mSin(2))?

I get $\displaystyle \frac{1}{3}e^m(5-{\color{red}4}e^m)$
SEE HERE

 

[Dr.Peterson]

I don't understand how to get from the top step to the next. Am I missing some ruler with Euler's number?

When I try to work through it using order of operations I end up with -1/3e^m + 1/3e^2m

Wouldn't 1/3e^2m end up as 1/3(Cos(2)+mSin(2))?

Can you show the steps you took, so we can see what mistakes you made?

I would do something like this:

[MATH]\left(1-\frac{2}{3}\left(e^m - 1\right)\right)e^m = \left(1-\left(\frac{2}{3}e^m - \frac{2}{3}\cdot1\right)\right)e^m = \left(\frac{3}{3}-\frac{2}{3}e^m + \frac{2}{3}\right)e^m = \left(\frac{5}{3}-\frac{2}{3}e^m\right)e^m = \frac{1}{3}\left(5-2e^m\right)e^m[/MATH]
I see that pka interpreted your question as asking for the derivative of the given function, whereas this is just one step in algebraically simplifying the derivative of some function you didn't show.

 

[raveyp]

Can you show the steps you took, so we can see what mistakes you made?

I would do something like this:

[MATH]\left(1-\frac{2}{3}\left(e^m - 1\right)\right)e^m = \left(1-\left(\frac{2}{3}e^m - \frac{2}{3}\cdot1\right)\right)e^m = \left(\frac{3}{3}-\frac{2}{3}e^m + \frac{2}{3}\right)e^m = \left(\frac{5}{3}-\frac{2}{3}e^m\right)e^m = \frac{1}{3}\left(5-2e^m\right)e^m[/MATH]
I see that pka interpreted your question as asking for the derivative of the given function, whereas this is just one step in algebraically simplifying the derivative of some function you didn't show.

This is great thanks, Dr. P

I think where I went wrong was by not starting with multiplying out the inner brackets first i.e. -2/3(e^m-1)

I overlooked that simple OoO, got confused and started thinking I was missing more complicated.