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ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

Multivariable Differentiation

Hi can someone please show full working on how to get the first derivative(with respect to x) of Z=arctan(2xy/(x^2-y^2)) Thanks!! By the way this is to help me solve question 60 of chapter 14.5 in Anton Calculus 7th edition (can refer to this if you have it)    Cheers guys/gals!

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

sameer mishra

Member

Registered: 2010-08-27

Posts: 64

Re: Multivariable Differentiation

hello my friend in my openion arctan(x)=tan^-1(x)
tan inverse x
above function can be written as arctan[2(y/x)/(1-y^2/x^2)]
now let y/x=tanb
F(x)=arctan[2tanb/(1-tan^ 2b)]
F(x)=arctan[tan2b]
F(x)=2b
F(x)'=2*b'
where b=tan inverse y/x

Bob

Administrator

Registered: 2010-06-20

Posts: 10,149

Re: Multivariable Differentiation

Hi ilovealgebra

I'VE CORRECTED THIS HAVING SEEN THE SUBSEQUENT POSTS. THANKS GUYS.

I've not got access to that book.  I'm wondering whether 'y' is to be treated as a variable with respect to 'x' as this will make a difference when the function of x and y is differentiated.

I'll start by treating y as a constant and then show what to do if it isn't.

Re-write as

Now differentiate each side with respect to x

...line A

but

so

re-arrange to give

which simplifies to

If 'y' is treated as a function of x then line A becomes

The rest follows similarly.

Bob

Last edited by Bob (2010-11-21 10:18:57)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

soroban

Member

Registered: 2007-03-09

Posts: 452

Re: Multivariable Differentiation

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Multivariable Differentiation

Hi ilovealgebra;

What a great username! I hope you love polar coordinates, cause that's where we are heading. Polar, and I do not mean bears!

Convert to polar.

A nifty substitution.

Again using polar coordinates.

Z has been greatly simplified by our polar friends.

Use these 2 beautiful formulas and solve for partial of x in either one.

Doing the partial differentiations above and solving for partial of x.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

Re: Multivariable Differentiation

Thanks everyone for your replies, I can now do the rest of the question

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Multivariable Differentiation

Hi ilovealgebra;

What rest of the question? You mean there is more!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

Re: Multivariable Differentiation

The full question: Show that the function

satisfies Laplace's equation

Then make the substitution

and show that the resulting function r and theta satisfies the polar form of Laplace's equation

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Multivariable Differentiation

Hi;

You have this:

You differentiate with respect to x to get the second partial derivative. Then you will need the second partial derivative with respect to y.
Then plug in to Laplace's equation.

I think the polar one will be easier!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ilovealgebra

Member

Registered: 2006-10-02

Posts: 40

Re: Multivariable Differentiation

Yeah It's a beast of a question but managed to get it out after seeing the working for computing for dz/dx

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"...nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called into question (much less condemned) upon the testimony of biblical passages."
-Galileo Galilei

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Multivariable Differentiation

Hi;

How did you show that it satisfies the Laplace equation? Oh, wait I am seeing it now! Made a typo error!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.