Problem Set 6

1. Easy
Using the limit definition of the derivative, show that each of the following functions are differentiable on their domain.
1. $$f: \mathbb R^2 \to \mathbb R, f(x,y) = x-y$$
2. $$f: \mathbb R^2 \to \mathbb R, f(x,y) = x^2+y$$
3. $$f: \mathbb R^2 \to \mathbb R, f(x,y) = x^n, n \in \mathbb N$$
4. $$f: \mathbb R^3\setminus\{(0,0)\} \to \mathbb R, f(x,y) = \frac{1}{x^2+y^2}$$
5. $$f: \mathbb R^3\setminus\{z=0\} \to \mathbb R, f(x,y) = \frac{xy}{z}$$

2. Easy
Consider the function $$f(x,y)$$ given in Equation (2.3) of the course notes. Show that this function is everywhere differentiable but is not $$C^1$$.

3. Easy
Let $$f: \mathbb R^n \to\mathbb R$$. Show that by using a particular choice of $$\mathbf u$$, one can recover the partial derivatives from the directional derivatives; that is, for each $$i \in \{1,\ldots,n\}$$ there exists $$\mathbf u$$ such that $$\partial_{\mathbf u} f(\mathbf a) = \partial_i f(\mathbf a)$$.

4. Easy
Let $$f: \mathbb R^2 \to \mathbb R, f(x,y) = xy^2 + e^{xy}$$, and set
$$\xi(x,y) = \partial_x f(x,y), \qquad \eta(x,y) = \partial_y f(x,y).$$
Compute $$\nabla \xi, \nabla \eta$$. Do you see a relationship between the derivatives of $$\xi$$ and $$\eta$$?

5. Easy
For each given function $$f$$, determine $$\nabla f$$ and give the domain on which $$f$$ is differentiable.
1. $$f(x,y) = \sin(y) e^{x^2+y^2}$$
2. $$f(x,y,z) = xy + xz + yz$$
3. $$f(x,y) = \frac1{xy-1}$$
4. $$f(x,y,z) = \frac{1}{(x^2+y^2+z^2)^{3/2}}$$

6. Easy
Let $$f,g: \mathbb R \to \mathbb R^n$$ and $$\varphi: \mathbb R \to \mathbb R$$ be differentiable functions. Show that
1. $$(\varphi f)' = \varphi ' f + \varphi f'$$
2. $$(f \cdot g)' = f' \cdot g + f \cdot g'$$
3. $$(f \times g)' = f' \times g + f \times g'$$ [when $$n=3$$].

7. Easy
Compute the partial derivatives $$\partial_{i}f$$, and the total derivative $$Df$$ for each of the following functions $$f:\mathbb{R}^{n}\to\mathbb{R}^{k}$$:
1. $$f(x,y,z) = x^{y}$$
2. $$f(x,y) = x^{y}$$
3. $$f(x,y,z) = (x^{y},z)$$
4. $$f(x,y) = \sin(x\sin(y))$$
5. $$f(x,y,z) = (x+y)^{z}$$
6. $$f(x,y,z) = (\log\left(x^2+y^2+z^2\right),xyz)$$
7. $$f(x,y)=\sin(xy)$$

8. Medium
Find an example of a function for which the directional derivatives in every direction exist, but the function is not differentiable.

9. Medium
Show that $$f:\mathbb{R}^{2}\to\mathbb{R}$$, $$f(x,y) = \sqrt{\vert xy\vert}$$ is not differentiable at $$(x,y) = (0,0)$$.

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