Find the derivative (Implicit)

[hoch449]

Homework Statement

Find \frac{\partial\theta}{\partial y}

z=rcos\theta
x=rsin\theta\cos\phi
y=rsin\theta\sin\phi
r^2=x^2 + y^2 + z^2

The Attempt at a Solution

We know cos\theta=\frac{z}{r}=\frac{z}{\sqrt{x^2 + y^2 + z^2}}

So implicit differentiation says to differentiate both sides with respect to y and this is where I begin to run into trouble.

Please be very specific when you try to explain how this is done lol.

Thanks!

 

[rock.freak667]

Do you know how to do implicit differentiation? Do you know how find the partial derivative of a function with respect to a variable?

 

[hoch449]

An earlier part to this question was to find \frac{\partial r}{\partial y} and I solved it correctly.

Here is how I did it.

r^2= x^2 + y^2 + z^2

\frac{d}{dy}r^2= \frac{d}{dy}y^2

\frac{d}{dr}r^2\frac{dr}{dy}=2y

2r\frac{dr}{dy}=2y

so therefore \frac{\partial r}{\partial y}= \frac{y}{r}

I am just having some difficulty with the next part of the question.

 

[slider142]

The Attempt at a Solution

We know cos\theta=\frac{z}{r}=\frac{z}{\sqrt{x^2 + y^2 + z^2}}

So implicit differentiation says to differentiate both sides with respect to y and this is where I begin to run into trouble.

In more specific terms, differentiate both sides of the equation with respect to y keeping all variables other than y and those that are explicit functions of y constant.
Where specifically did you run into trouble?

 

[hoch449]

The Right Hand Side of the equation gives me the difficulty. I am sure I am making an elementary mistake.

\frac{\partial\theta}{\partial y}:

\frac{d}{dy}cos\theta=\frac{d}{dy}(\frac{z}{r})

\frac{d}{d\theta}(cos\theta)\frac{d\theta}{dy}=\frac{d}{dr}r^-1\frac{dr}{dy}

-sin\theta\frac{d\theta}{dy}=-r^-2\frac{dr}{dy}

I have a feeling that I have already made a mistake with the right side...