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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