Question

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

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

Short answer

The value of provided equation is ${\left(\frac{\partial \mathrm{z}}{\partial \mathrm{x}}\right)}_{\mathrm{\theta }}=2x\left(1+2{\tan }^2\theta \right)$.

Step-by-step solution

Given data

The given data are listed below

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

 Step 2: 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

From equation 1 and 2,

$$\begin{gathered} \frac{y}{x}=\tan \theta \\ y=x\tan \theta \end{gathered}$$

Substitute the value of y in equation 1.

$$\begin{gathered} z=x^2+2{\left(x\tan \theta \right)}^2 \\ z=x^2\left(1+2{\tan }^2\theta \right) \end{gathered}$$

Take the partial derivative of zwith respect to x,

role="math" localid="1659009652553" $$\begin{gathered} {\left(\frac{\partial z}{\partial x}\right)}_{\theta }=\frac{\partial }{\partial x}\left(x^2\left(1+2{\tan }^2\theta \right)\right) \\ {\left(\frac{\partial z}{\partial x}\right)}_{\theta }=2x\left(1+2{\tan }^2\theta \right) \end{gathered}$$

Therefore,${\left(\frac{\partial z}{\partial x}\right)}_{\theta }=2x\left(1+2{\tan }^2\theta \right)$.