Question

If $\mathit{z}\mathbf{=}\mathit{I}\mathit{n}\sqrt{{\mathbf{u}}^{\mathbf{2}}\mathbf{+}{\mathbf{v}}^{\mathbf{2}}\mathbf{+}{\mathbf{w}}^{\mathbf{2}}}$,find$\mathbf{\partial }\mathit{z}\mathbf{/}\mathbf{\partial }\mathit{u}\mathbf{,}\mathbf{\partial }\mathit{z}\mathbf{/}\mathbf{\partial }\mathit{v}\mathbf{,}\mathbf{\partial }\mathit{z}\mathbf{/}\mathit{w}$

Short answer

The value of provided equation is$\frac{\partial z}{\partial u}=\frac{u}{u^2+v^2+w^2}\frac{\delta z}{\delta v}=\frac{v}{u^2+v^2+w^2}$ , and $\frac{\delta z}{\delta w}=\frac{w}{u^2+v^2+w^2}$.

Step-by-step solution

Explanation of solution

In this question, it is provided that $\mathrm{z}=\mathrm{In}\sqrt{{\mathrm{u}}^2+{\mathrm{v}}^2+{\mathrm{w}}^2}$and

Find$\partial z/\partial u,\partial z/\partial v,\partial z/w$

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

It is provided that$z=\sqrt{u^2+v^2+W^2}$

Implies that,

$z=\frac{1}{2}In\left(u^2+v^2+w^2\right)$

Now, Take partial derivative of z with respect to u,

$$\begin{gathered} \frac{\delta z}{\delta u}=\frac{\delta }{\delta u}\left[\frac{1}{2}In\left(u^2+v^2+w^2\right)\right] \\ =\frac{1}{2}\frac{\delta }{\delta u}In\left(\frac{1}{u^2+v^2+w^2}\right)\times \left(2u\right) \end{gathered}$$

So,

$\frac{\partial z}{\partial u}=\frac{u}{u^2+v^2+w^2}$

Now,

$$\begin{gathered} \frac{\delta z}{\delta u}=\frac{\delta }{\delta u}\left[\frac{1}{2}In\left(u^2+v^2+w^2\right)\right] \\ =\frac{1}{2}.\frac{\delta }{\delta u}.In.\left(u^2+v^2+w^2\right).2v \\ =\frac{1}{2}\left(\frac{1}{u^2+v^2+w^2}\right).2v \end{gathered}$$

So,

$$\begin{gathered} \frac{\mathit{\delta }\mathit{z}}{\mathit{\delta }\mathit{v}}\mathit{=}\frac{\mathit{v}}{{\mathit{u}}^{\mathit{2}}\mathit{+}{\mathit{v}}^{\mathit{2}}\mathit{+}{\mathit{w}}^{\mathit{2}}} \\ \mathrm{Similarly},\frac{\mathrm{\delta z}}{\mathrm{\delta w}}=\frac{\mathit{w}}{{\mathit{u}}^{\mathit{2}}\mathit{+}{\mathit{v}}^{\mathit{2}}\mathit{+}{\mathit{w}}^{\mathit{2}}} \end{gathered}$$