# Partial differentiation question?

1. Jan 16, 2014

### applestrudle

1. The problem statement, all variables and given/known data

z = x^2 +y^2

x = rcosθ

y = rsinθ

find partial z over partial x at constant theta

2. Relevant equations

z = x^2 +y^2

x = rcosθ

y = rsinθ

3. The attempt at a solution

z = 1 + r^2(sinθ)^2

dz/dx = dz/dr . dr/dx

= 2(sinθ)^2r/cosθ

= 2tanθ^2x

the book says 2x[1+2(tanθ)^2]

2. Jan 16, 2014

### tiny-tim

hi applestrudle!

(have a curly d: ∂ and try using the X2 button just above the Reply box )
∂/∂x is ambiguous unless you know what the other variables are

it always means that you differentiate wrt x keeping the other variables constant (that's why you need to know what they are!)

in this case, the question tells you the other variable is θ, so first you need to find z(x,θ), ie to write z as a function of x and θ

(hmm … i don't get the result the book gets )

3. Jan 16, 2014

### vanhees71

Here, I'd write $z$ as function of $x$ and $\theta$. Then it's easy to take the partial derivative. Obviously we have
$$z=r^2=\frac{x^2}{\cos^2 \theta}.$$
Then you can take the partial derivative wrt. $x$ and fixed $\theta$ easily, but what you quoted as solution of the book is obviously wrong (the factor 2 in front of $\tan^2 \theta$ should not be there).

Last edited: Jan 16, 2014