• 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

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

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