derivatives

Hi,

I study electronic engineering taught in english in another country,so there's a lot of maths involved.Sadly,I must say that the teaching quality should improve a lot,but I don't have what to do,so I'm self-teaching myself maths.

I've started reading some paper about derivatives and then did lots of exercises,but now I found something new (and probably very simple),but I don't know how to start it.
Could someone please explain me the following exercise or link me some material that might help.

Thank you.

The exercise you have included is concerned with partial derivatives. These are not really much different to ordinary derivatives when calculating them, but things can be more complicated sometimes. I will assume that you are fine with usual differentiation (i.e., differentiating a function of one variable).

Now if we have a function

u(x,y)

, then as it depends on two variables, we could ask about how

u

changes with respect to one of these. If we are interested in how

u

changes with

x

, then we should ignore the effects of

y

- to do this, we simply pretend it is a constant.

So suppose we have

u(x,y)=x^2+4xy+3y

. To differentiate with respect to

x

, simply pretend

y

is a constant. You should find

\dfrac{\partial u}{\partial x} = 2x+4y

. The funny looking d's,

\partial

are called partials d's, and remind us that

u

as a function depends on more than just the variable we have differentiated with respect to.

It is possible to differentiate with respect to both variables, in some order. We could differentiate the above answer with respect to y this time;

\dfrac{\partial^2 u}{\partial y\partial x} = 4

. Notice that doing x first, and then y, we write this as

\dfrac{\partial^2}{\partial y \partial x}u

. The reason they appear in that order is we could see it as

\dfrac{\partial}{\partial y}(\dfrac{\partial}{\partial x}u)

. So the order the derivatives were done is right-to-left.

Lastly, partial derivatives obey similar rules (like the product rule and chain rule) as the usual derivative. The chain rule is slightly more complicated, because the extra variables mean we may have to add different terms together. The chain rule for two variables is as follows:

\dfrac{\partial u(\xi(x,y),\eta(x,y))}{ \partial x} = \dfrac{\partial u}{\partial \xi}\dfrac{\partial \xi}{\partial x}+\dfrac{\partial u}{\partial \eta}\dfrac{\partial \eta}{\partial x}

This rule (along with the product rule, which is the same as for ordinary differentiation) is what you need to do the exercise you posted. If you have any questions, please feel free to ask.

Reply 2

laur21

OP

2

You explained very well,even a child could understand..The only thing which was unfamiliar is

Thank you and I believe you should become a math teacher ^_^

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Reply 3

TeeEm

19

Original post by laur21

Hi,

I study electronic engineering taught in english in another country,so there's a lot of maths involved.Sadly,I must say that the teaching quality should improve a lot,but I don't have what to do,so I'm self-teaching myself maths.

I've started reading some paper about derivatives and then did lots of exercises,but now I found something new (and probably very simple),but I don't know how to start it.
Could someone please explain me the following exercise or link me some material that might help.

Thank you.

http://madasmaths.com/archive/maths_booklets/advanced_topics/partial_differentiation.pdf

out of interest which country?