Kelton Bailey

2022-09-03

Partial differentiation with log function.

Can someone please help as I am stuck,

I need to show that

$$\varphi =\frac{k}{2\pi}\mathrm{log}({x}^{2}+{y}^{2}{)}^{1/2}$$

satisfies Laplaces equation, however I cannot seem to differentiate this function. Note k is a constant.

How do I go about partially differentiating

$$\mathrm{log}(\sqrt{{x}^{2}+{y}^{2}})$$

I was thinking, using chain rule, just call

$$\sqrt{{x}^{2}+{y}^{2}}=r$$

so

$$\frac{1}{r}\mathrm{log}r+\mathrm{log}r$$

Any help is appreciated.

Can someone please help as I am stuck,

I need to show that

$$\varphi =\frac{k}{2\pi}\mathrm{log}({x}^{2}+{y}^{2}{)}^{1/2}$$

satisfies Laplaces equation, however I cannot seem to differentiate this function. Note k is a constant.

How do I go about partially differentiating

$$\mathrm{log}(\sqrt{{x}^{2}+{y}^{2}})$$

I was thinking, using chain rule, just call

$$\sqrt{{x}^{2}+{y}^{2}}=r$$

so

$$\frac{1}{r}\mathrm{log}r+\mathrm{log}r$$

Any help is appreciated.

Cullen Kelly

Beginner2022-09-04Added 7 answers

I would think that you would immediately use $log(\sqrt{{x}^{2}+{y}^{2}})=\frac{1}{2}log({x}^{2}+{y}^{2})$. That will simplify your problem.

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