First Order Partial Derivatives of a Function

[Joeda]

Find the first order partial derivatives of the function x = f(x,y) at the point (4,3) where:
$$f(x,y)=ln|(x+√(x^2+y^2))/(x-√(x^2+y^2))|$$

I understand the method of partial derivatives and implementing the given point values once the partial derivatives are found, however I am having trouble trying to simplify the equation so that the partial derivatives can be found.

I have used the log rule to simplify the function but I think it can be simplified further but am stuck. So far I've got:

$$In(x+√(x^2+y^2)-In(x-√(x^2-y^2)$$

Do I then multiply out the In by what's in the brackets? Cause that doesn't look right when I work through the problem. Looking at similar problems I am guessing the equation being simplified would be
$$f(x,y) =x+√(x^2+y^2)$$
but how do I get there?

 

[voko]

$$\ln (x+\sqrt(x^2+y^2)-\ln(x-\sqrt(x^2-y^2)$$

I do not see how you got the minus in the second radical. It was not in the original expression.

What you could do is divide the numerator and denominator of the original expression under log by x before you convert that to a difference of logs.

Or you could just bite the bullet and compute partial derivatives of what you already have (after you fix the minus sign).

 

[Joeda]

Thanks Voko I will give that a try.
Yes sorry the second minus sign is a typo.

 

[Joeda]

ok hows this look?

Dividing both numerator and denominator by x

$$f(x,y)= In|(√(x^2+y^2))/(√(x^2+y^2))|$$

Simplifying using Log Rules

$$f(x,y)= In|(√(x^2+y^2))|-In|(√(x^2+y^2))|$$

$$f(x,y)= (1/2)In|((x^2+y^2))|-(1/2)In|((x^2+y^2))|$$

$$f(x) = (1/2)(1/(x^2+y^2)-(1/2)(1/(x^2+y^2)$$

$$f(x) = (1/2)(1/(x^2+y^2)(2x)-(1/2)(1/(x^2+y^2)(2x)$$

$$f(x) = (x/(x^2+y^2)-(x/(x^2+y^2)$$ (wouldnt this just equal zero?)

$$f(y) = (1/2)(1/(x^2+y^2)(2y)-(1/2)(1/(x^2+y^2)(2y)$$

$$f(y) = (y/(x^2+y^2)-(y/(x^2+y^2)$$ (wouldnt this just equal zero?)

 

[Dick]

ok hows this look?

Dividing both numerator and denominator by x

$$f(x,y)= In|(√(x^2+y^2))/(√(x^2+y^2))|$$

Simplifying using Log Rules

$$f(x,y)= In|(√(x^2+y^2))|-In|(√(x^2+y^2))|$$

$$f(x,y)= (1/2)In|((x^2+y^2))|-(1/2)In|((x^2+y^2))|$$

$$f(x) = (1/2)(1/(x^2+y^2)-(1/2)(1/(x^2+y^2)$$

$$f(x) = (1/2)(1/(x^2+y^2)(2x)-(1/2)(1/(x^2+y^2)(2x)$$

$$f(x) = (x/(x^2+y^2)-(x/(x^2+y^2)$$ (wouldnt this just equal zero?)

$$f(y) = (1/2)(1/(x^2+y^2)(2y)-(1/2)(1/(x^2+y^2)(2y)$$

$$f(y) = (y/(x^2+y^2)-(y/(x^2+y^2)$$ (wouldnt this just equal zero?)

Doesn't look too good yet. Dividing numerator and denominator by x doesn't make the x just disappear. And even if you did it correctly it doesn't help much. Just change it into the difference of logs and differentiate each part.

 

[Joeda]

ok I am really starting to get confused here. How about this?

$$f(x,y)=In|x+√(x^2+y^2)|-In|x-√(x^2+y^2)|$$

$$f(x,y)=(1/2)In|x+(x^2+y^2)|-(1/2)In|x-(x^2+y^2)|$$

$$f(x)=(1/x)(1/2)(1/(x+x^2+y^2))(1+2x+y^2)-(1/x)(1/2)(1/(x-x^2+y^2))(1-2x+y^2)$$

$$f(x)=(1/x)+(0.5+x+0.5y^2)/(x+x^2+y^2)-(1/x)(0.5-x-0.5y^2/(x-x^2+y^2)$$

 

[voko]

$$\frac {x + \sqrt{x^2 + y^2}} {x} = \frac {x}{x} + \frac {\sqrt{x^2 + y^2}} {\sqrt {x^2}} = 1 + \sqrt {\frac {x^2} {x^2} + \frac {y^2} {x^2}} = 1 + \sqrt {1 + \frac {y^2}{x^2}}$$

 

[HallsofIvy]

Find the first order partial derivatives of the function x = f(x,y) at the point (4,3) where:
$$f(x,y)=ln|(x+√(x^2+y^2))/(x-√(x^2+y^2))|$$

I understand the method of partial derivatives and implementing the given point values once the partial derivatives are found, however I am having trouble trying to simplify the equation so that the partial derivatives can be found.

I have used the log rule to simplify the function but I think it can be simplified further but am stuck. So far I've got:

$$In(x+√(x^2+y^2)-In(x-√(x^2-y^2)$$

That should be "ln" not "In":
$$ln(x+\sqrt{x^2+ y^2}- ln(x- \sqrt{x^2+ y^2}$$
It cannot be simplified further because there is no good "ln(a+ b)" identity.

Do I then multiply out the In by what's in the brackets? Cause that doesn't look right when I work through the problem. Looking at similar problems I am guessing the equation being simplified would be
$$f(x,y) =x+√(x^2+y^2)$$
but how do I get there?

I wouldn't multiply. The derivative of ln(x), with respect to x, is 1/x so you will have
$$\frac{1}{x+ \sqrt{x^2+ y^2}}$$
times the derivative of the quantity $x+ (x^2+ y^2)^{1/2}$ with respect to either x or y plus
$$\frac{1}{x-\sqrt{x^2+ y^2}}$$
times the derivative of $x- (x^2+ y^2)^{1/2}$.

 

[Joeda]

Thanks everyone for all your help. So...

$$f(x,y)=ln|\frac{x+\sqrt{x^2+y^2}}{x-\sqrt{x^2+y^2}}|$$

$$f(x,y)=ln|(x+\sqrt{x^2+y^2})| - ln|(x-\sqrt{x^2+y^2})|$$

$$f(x)=\frac{1}{x+\sqrt{x^2+y^2}} . \frac{3}{2}(2x+y^2)^{-1/2} - \frac{1}{x-\sqrt{x^2+y^2}} . \frac{1}{2}(2x+y^2)^{-1/2}$$

 

[voko]

That's not correct. $$\frac \partial {\partial x} \ln f(x, y) = \frac 1 {f(x, y)} \frac {\partial f} {\partial x}$$ You got $\frac 1 {f(x, y)}$ right, but $\frac {\partial f} {\partial x}$ is wrong.

 

[Joeda]

Is the power outside of the brackets not required to be derived?
Also if x & y are inside brackets when taking the partial derivative of x, isn't the derivative of x taken and y is just a constant?
ie. $$f(x) = {(x^2+y^2)}$$
$$= 2x + y^2$$
whereas if it was
$$f(x) = {x^2+y^2}$$
$$= 2x$$

 

[voko]

When you have $$f(x, y) = [g(x, y)]^a$$ then $$\frac \partial {\partial x} f(x, y) = a[g(x, y)]^{a - 1}\frac \partial {\partial x} g(x, y)$$

 

[Joeda]

voko I am lost now.

 

[Joeda]

So does
$$x+(x^2+y^2)^{1/2}$$
equal
$$x+\frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$ ?

 

[voko]

They are not equal, but your question was probably is the latter the derivative of the former? Also not, but pretty close. What is the derivative of x with respect to x?

 

[Joeda]

Derivative of x with respect to x is 1. So its

$$1+\frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$
$$\frac{3}{2}(x^2+y^2)^{-1/2}(2x)$$
Please tell me this is correct!

 

[voko]

Derivative of x with respect to x is 1. So its

$$\frac{3}{2}(x^2+y^2)^{-1/2}(2x)$$

Please tell me this is correct!

Unfortunately, no. Where did 3 come from? This part was correct in the previous message.

So then for the question above, (2nd part) the Derivative with respect to y for

$$f(y)= x-(x^2+y^2)^{1/2}$$ is

$$= x-\frac{1}{2}(x^2+y^2)^{-1/2}(2y)$$

What is the derivative of x with respect to y?

 

[Joeda]

Well the 3 came from the derivative of x with respect to x = 1
1+1\2=3/2

The derivative of x with respect to y is zero?

 

[voko]

Well the 3 came from the derivative of x with respect to x = 1
1+1\2=3/2

NO, NO, NO! You have $a + b c$. This is not equal to $(a + b)c$.

The derivative of x with respect to y is zero?

Correct.

 

[Joeda]

Sorry Sir once again I am lost :)

If I have $$(x+y^2)^3$$
using the chain rule I get
$$\frac{df}{dx}=3(x+y^2)^{2}$$ as derivative of x = 1 so this could be written as
$$\frac{df}{dx}=3(x+y^2)^{2} *(1)$$
and
$$\frac{df}{dy}=3(x+y^2)^{2}*(2y)$$
$$\frac{df}{dy}=6y(x+y^2)^{2}$$

So then shouldnt:

$$x+(x^2+y^2)^{1/2}$$
$$f(x) = 1+ \frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$
$$f(x) = \frac{3}{2}(x^2+y^2)^{-1/2}(2x)$$

 

[voko]

$$x+(x^2+y^2)^{1/2}$$
$$f(x) = 1+ \frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$

Correct.

$$f(x) = \frac{3}{2}(x^2+y^2)^{-1/2}(2x)$$

Incorrect. This is a very basic mistake, which has nothing to do with derivatives. This is just algebra, not calculus. Let $a = 1$, $b = \frac 1 2$ and $c = (x^2+y^2)^{-1/2}(2x)$.

Then the first expression is $a = bc$ and the second is $(a + b)c$. These are not equal, because $(a + b)c = ac + bc \ne a + bc$, except when $c = 1$, which is definitely not the case here. The correct expression is $$1+ \frac{1}{2}(x^2+y^2)^{-1/2}(2x) = 1+ (x^2+y^2)^{-1/2}x$$

 

[Joeda]

Wow I can't believe I didnt see that!
Amazing how you can get caught up in the more difficult aspects of a question and miss the basics!
Thankyou Voko for your patience its very much appreciated.

 

[Joeda]

The correct expression is $$1+ \frac{1}{2}(x^2+y^2)^{-1/2}(2x) = 1+ (x^2+y^2)^{-1/2}x$$

So can this expression be simplified further by multiplying the x term outside the brackets by the terms inside the brackets, or should I leave these terms as is and just plug in my values for x & y?
I would think you can't but just wanted to clarify.

 

[voko]

After you differentiate the entire log, you should have two very similar terms for each derivative. Those could be combined into something a bit more compact.

 

[Joeda]

So I get
$$f(x)=\frac{1}{x+\sqrt{x^2+y^2}} . 1+(x^2+y^2)^{-1/2}(x) - \frac{1}{x-\sqrt{x^2+y^2}} . 1-(x^2+y^2)^{-1/2}(-x)$$

 

[voko]

So I get
$$f(x)=\frac{1}{x+\sqrt{x^2+y^2}} . 1+(x^2+y^2)^{-1/2}(x) - \frac{1}{x-\sqrt{x^2+y^2}} . 1-(x^2+y^2)^{-1/2}(-x)$$

That should be written as $$f(x)=\frac{1}{x+\sqrt{x^2+y^2}} [1+(x^2+y^2)^{-1/2}(x)] - \frac{1}{x-\sqrt{x^2+y^2}} [1-(x^2+y^2)^{-1/2}(-x)]$$

Now, do you see that both terms in square brackets are equal so that you could use that as a common factor?

 

[voko]

No, wait. How did you get (-x) in the second term? That does not seem correct.

 

[Joeda]

That should be written as $$f(x)=\frac{1}{x+\sqrt{x^2+y^2}} [1+(x^2+y^2)^{-1/2}(x)] - \frac{1}{x-\sqrt{x^2+y^2}} [1-(x^2+y^2)^{-1/2}(-x)]$$

Now, do you see that both terms in square brackets are equal so that you could use that as a common factor?

So we get

$$f(x)= [1+(x^2+y^2)^{-1/2}(x)] [\frac{1}{x+\sqrt{x^2+y^2}} - \frac{1}{x-\sqrt{x^2+y^2}}]$$

That doesn't look right as we had (x) and (-x) in the above equation.

 

[Joeda]

As I had

$$f(x)= x+(x^2+y^2)^{1/2}$$
$$f(x)= 1+\frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$
$$f(x)= 1+(x^2+y^2)^{-1/2}(x)$$

and

$$f(x)= x-(x^2+y^2)^{1/2}$$
$$f(x)= 1-\frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$
$$f(x)= 1-(x^2+y^2)^{-1/2}(-x)$$

As $$-\frac{1}{2} * 2x = -x$$

 

[voko]

$$f(x)= 1-\frac{1}{2}(x^2+y^2)^{-1/2}(2x)$$
$$f(x)= 1-(x^2+y^2)^{-1/2}(-x)$$

As $$-\frac{1}{2} * 2x = -x$$

But you got two minus signs where you only had one. That's not correct. There should not be (-x), it should just be (x).

 

[Joeda]

ok once again thanks Voko you have been brilliant!

$$f(x)= [1+(x^2+y^2)^{-1/2}(x)] [\frac{1}{x+\sqrt{x^2+y^2}} - \frac{1}{x-\sqrt{x^2+y^2}}]$$

So this is written correctly then?

 

[voko]

Again, the second term is incorrect. You cannot go from (-a)(b) to -(-ab), but that's what you insist on doing. (-a)(b) = -(ab).