Question

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

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

Short answer

The value of provide equation is ${\left(\frac{\partial z}{\partial \theta }\right)}_x=4r^2\sin \theta \cos \theta$.

Step-by-step solution

Given data

The given data are listed below

$z=x^2+2y^2$ ...(1)

$x=r\cos \theta$ ...(2)

$y=r\sin \theta$ ...(3)

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 3,

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

Now, take the partial derivative with respect to in above equation,

role="math" localid="1659091215113" $$\begin{gathered} {\left(\frac{\partial z}{\partial \theta }\right)}_x=\frac{\partial }{\partial \theta }\left(x^2+2r^2{\sin }^2\theta \right) \\ {\left(\frac{\partial z}{\partial \theta }\right)}_x=4r^2\sin \theta \cos \theta \end{gathered}$$

Hence,${\left(\frac{\partial z}{\partial \theta }\right)}_x=4r^2\sin \theta \cos \theta$