Partial Derivative Explanation

[Desii]

Hello,
Could anyone please explain me the steps in these pictures.
I do not understand the second step.

http://imgur.com/AvVbPu5,Ust2Zpx#0

Second one:
Third step ( i don't understand)
http://imgur.com/AvVbPu5,Ust2Zpx#1

If anyone can give me detail explanation, i would really appreciate it.

 

[fzero]

Hello,
Could anyone please explain me the steps in these pictures.
I do not understand the second step.

http://imgur.com/AvVbPu5,Ust2Zpx#0

Second one:
Third step and last step ( i don't understand)
http://imgur.com/AvVbPu5,Ust2Zpx#1

If anyone can give me detail explanation, i would really appreciate it.

Have you tried using the chain rule? The text is omitting some algebra, but you should be able to fill it in if you just try to do the computation your own way.

 

[Fredrik]

The last step in the second one is obvious, isn't it? (It's essentially just A+B-A=0). I could tell you exactly what to do with the rest, but I'm not sure this would help your understanding at all, so I will start by only giving you some hints. Maybe the problem here isn't that you don't know how to apply the rules, but that you don't know why you can apply them?

You need the following rules:

Linearity: $(af+bg)'(x)=af'(x)+bg'(x)$.
Quotient rule: $\displaystyle \left(\frac f g\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}.$

In the first one, I would start with this very simple rewrite:
$$a=\frac{M-m}{M+m}g =\frac{Mg-mg}{M+m}.$$ Then you can just use the quotient rule (and linearity) to compute the partial derivatives. (You didn't include enough information for me to see why the last line is true).

The second one is similar.

 

[Desii]

The last step in the second one is obvious, isn't it? (It's essentially just A+B-A=0). I could tell you exactly what to do with the rest, but I'm not sure this would help your understanding at all, so I will start by only giving you some hints. Maybe the problem here isn't that you don't know how to apply the rules, but that you don't know why you can apply them?

You need the following rules:

Linearity: $(af+bg)'(x)=af'(x)+bg'(x)$.
Quotient rule: $\displaystyle \left(\frac f g\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}.$

In the first one, I would start with this very simple rewrite:
$$a=\frac{M-m}{M+m}g =\frac{Mg-mg}{M+m}.$$ Then you can just use the quotient rule (and linearity) to compute the partial derivatives. (You didn't include enough information for me to see why the last line is true).

The second one is similar.

Hi,
Thank you very much for the help, I'm used to solving it with numbers not in problems. It makes all sense now. Once again thank you