Chapter 4: Q3P (page 190)

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

Short Answer

Expert verified

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

Step by step solution

Explanation of solution

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

$z = \frac{1}{2} I n (u^{2} + v^{2} + w^{2})$

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

$\frac{δ z}{δ u} = \frac{δ}{δ u} [\frac{1}{2} I n (u^{2} + v^{2} + w^{2})] = \frac{1}{2} \frac{δ}{δ u} I n (\frac{1}{u^{2} + v^{2} + w^{2}}) \times (2 u)$

So,

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

Now,

$\frac{δ z}{δ u} = \frac{δ}{δ u} [\frac{1}{2} I n (u^{2} + v^{2} + w^{2})] = \frac{1}{2} . \frac{δ}{δ u} . I n . (u^{2} + v^{2} + w^{2}) . 2 v = \frac{1}{2} (\frac{1}{u^{2} + v^{2} + w^{2}}) . 2 v$

So,

$\frac{δ z}{δ v} = \frac{v}{u^{2} + v^{2} + w^{2}} Similarly, \frac{δz}{δw} = \frac{w}{u^{2} + v^{2} + w^{2}}$

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

Short Answer

Step by step solution

Explanation of solution

Partial differentiation

Calculation

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If z=Inu2+v2+w2,find∂z/∂u,∂z/∂v,∂z/w

Short Answer

Step by step solution

Explanation of solution

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One App. One Place for Learning.

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