• Title: Directional Derivative

  • Series: Multivariable Calculus

  • YouTube-Title: Multivariable Calculus 10 | Directional Derivative

  • Bright video: https://youtu.be/OxVYmBZqeBU

  • Dark video: https://youtu.be/9TOfdo4yem8

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

  • Thumbnail (dark): Download PNG

  • Subtitle on GitHub: mc10_sub_eng.srt missing

  • Timestamps (n/a)
  • Subtitle in English (n/a)
  • Quiz Content

    Q1: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and $\tilde{\mathbf{x}}, \mathbf{v} \in \mathbb{R}^n$. What is the definition of the directional derivative of $f$ along the vector $\mathbf{v}$ at the point $\tilde{\mathbf{x}}$.

    A1: $$ \lim_{h \rightarrow 0} \frac{ f( \tilde{\mathbf{x}} + h \mathbf{v} ) - f( \tilde{\mathbf{x}} )}{h} $$

    A2: $$ \lim_{h \rightarrow 0} \frac{ f( \tilde{\mathbf{x}} + h \mathbf{v} ) + f( \tilde{\mathbf{x}} )}{h} $$

    A3: $$ \lim_{h \rightarrow 0} \frac{ f( \tilde{\mathbf{x}} + h \mathbf{v} ) - f( \tilde{\mathbf{x}} + h )}{h} $$

    Q2: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and $\tilde{\mathbf{x}}, \mathbf{v} \in \mathbb{R}^n$. What is an alternative definition of the directional derivative of $f$ along the vector $\mathbf{v}$ at the point $\tilde{\mathbf{x}}$.

    A1: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} + t \mathbf{v} )|_{\mid t=0} $$

    A2: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} - t \mathbf{v} )|_{\mid t=1} $$

    A3: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} - \mathbf{v} )|_{\mid t=0} $$

    A4: $$ \frac{d}{dt} f( \tilde{\mathbf{x}} - t^2 \mathbf{v} )|_{\mid t=1} $$

    Q3: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be totally differentiable and $\tilde{\mathbf{x}}, \mathbf{v} \in \mathbb{R}^n$. How can the directional derivative be calculated?

    A1: $$ \langle \mathrm{grad} f (\tilde{\mathbf{x}}), \mathbf{v} \rangle $$

    A2: $$ \langle \mathrm{grad} f (\mathbf{v}), \tilde{\mathbf{x}} \rangle $$

    A3: $$ - \langle \mathrm{grad} f (\mathbf{v}), \tilde{\mathbf{x}} \rangle $$

    A4: $$ \langle \mathrm{grad} f (\tilde{\mathbf{x}}) - \mathbf{v}, \tilde{\mathbf{x}} \rangle $$

  • Back to overview page