Question

If,$\mathit{z}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}\mathbf{,}\mathit{x}\mathbf{=}\mathit{r}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{,}\mathbf{}\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathit{r}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }$, find the following partial derivatives.

${\left(\frac{\partial z}{\partial r}\right)}_{\mathbf{x}}$

Short answer

The value of provided equation is $\left(\frac{\partial z}{\partial r}\right),=4r$.

Step-by-step solution

Given data

The given data are listed below

$$\begin{gathered} z=x^2+2y^2 \\ x=r\cos \theta \\ \left(2\right) \\ y=r\sin \theta \end{gathered}$$

Partial differentiation

The procedure of calculating the partial derivative of a function is called partial differentiation. This method is used to get the partial derivative of a function with respect to one variable while keeping the other constant.

Calculation

Square the equation 2 and 3, then add.

$x^2+y^2=r^2$ ...(4)

From equation (1),

$$\begin{gathered} z=x^2+2(x^2+y^2)-2x^2 \\ z=-x^2+2r^2 \end{gathered}$$

Now, differentiate partially with respect to ,

$$\begin{gathered} {\left(\frac{\partial z}{\partial r}\right)}_x=\frac{\partial }{\partial r}\left[-x^2+2r^2\right] \\ {\left(\frac{\partial z}{\partial r}\right)}_x=4r \end{gathered}$$

Hence,${\left(\frac{\partial z}{\partial r}\right)}_x=4r$