Question

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

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

Short answer

The value of provided equation is, $\left(\frac{\partial z}{\partial x}\right)=-2x$.

Step-by-step solution

  Step 1: Given data

The given data are listed below

$$\begin{gathered} z=x^2+2y^2...\left(1\right) \\ x=r\cos \theta ....\left(2\right) \\ y=r\sin \theta .....\left(3\right) \end{gathered}$$

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),

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

Take the derivative ofwith respect to ,

role="math" localid="1659095459640" $$\begin{gathered} {\left(\frac{\partial z}{\partial x}\right)}_r=\frac{\partial }{\partial x}(-x^2+2r^2) \\ {\left(\frac{\partial z}{\partial x}\right)}_r=-2x \end{gathered}$$

Therefore, ${\left(\frac{\partial z}{\partial x}\right)}_r=-2x$.