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I need help with this: I need to find all points where this function is differentiable: $f:\mathbb{R}^{3}\rightarrow \mathbb{R}, \begin{pmatrix}x\\ y\\ z\end{pmatrix} \mapsto e^x \sin (z) + z \cos\bigl( \sqrt{x^2 + y^2 +1}\,\bigr).$

After that I must calculate the derivatives at that points.

This is a part of an exercise in my analysis class. It is the first part of a larger exercise but i need to understand this first to do the rest. I know so far that to calculate the derivatives i need to show that the function is continous partial differentiable. But I don't know how to find all these point and do it for them.

Can you give me a hint?

False Promise
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wolfffi
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  • Welcome to Math.SE: In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. – projectilemotion May 27 '17 at 14:18
  • Hi and thank you very much. This is a part of an exercise in my analysis class. It is the first part of a larger exercise but i need to understand this first to do the rest. I know so far that to calculate the derivatives i need to show that the function is continous partial differentiable. But i dont know how to find all these point and do it for them. – wolfffi May 27 '17 at 14:39
  • Does what I wrote below make sense? – Faraad Armwood May 27 '17 at 14:39

2 Answers2

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$$Df(\textbf{p}) = \begin{pmatrix} \frac{\partial}{\partial x}\bigr|_{\textbf{p}} \left(e^x \sin z + z \cos \sqrt{x^2 + y^2 +1} \right) \\ \frac{\partial}{\partial y}\bigr|_{\textbf{p}}\left(e^x \sin z + z \cos \sqrt{x^2 + y^2 +1} \right)\ \\\frac{\partial}{\partial z}\bigr|_{\textbf{p}}\left(e^x \sin z + z \cos \sqrt{x^2 + y^2 +1} \right)\end{pmatrix}$$

Since $x^2 + y^2 +1 \geq 0$, we have that $f$ is differentiable on $\mathbb{R}^3$. This is sufficient since when computing the partials of the second summand (in each entry), you will only get a convergent limit in the calculation of the derivative when $\sqrt{u} \geq 0$.

Faraad Armwood
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Thank you projectilemotion and Faraad,

i have uploaded an image. Is this ok so far? https://i.stack.imgur.com/nrW4H.png

wolfffi
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