• Title: Partial Derivatives

• Series: Multivariable Calculus

• YouTube-Title: Multivariable Calculus 4 | Partial Derivatives

• Bright video: https://youtu.be/kq5OCTDAXxw

• Dark video: https://youtu.be/bIPtcHmpRJ4

• Subtitle on GitHub: mc04_sub_eng.srt missing

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

Q1: Consider the function $f: \mathbb{R}^3 \rightarrow \mathbb{R}$. What is the correct definition for the notion: $f$ is partially differentiable with respect to $x_3$ at $\tilde{x}$?

A1: $$\lim_{h \rightarrow 0} \frac{f(\tilde{x})}{\tilde{x}}$$ exists.

A2: $$\lim_{h \rightarrow 0} \frac{f(\tilde{x}_1, \tilde{x}_2, \tilde{x}_3)}{h}$$ exists.

A3: $$\lim_{h \rightarrow 0} \frac{f(\tilde{x}_1, \tilde{x}_2, \tilde{x}_3 + h) - f(\tilde{x}) }{h}$$ exists.

A4: $$\lim_{h \rightarrow 0} \frac{f(\tilde{x}_1, \tilde{x}_2+h, \tilde{x}_3 + h) - f(\tilde{x}) }{h}$$ exists.

Q2: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $$f(x_1, x_2) = x_1 \cdot x_2$$ What is $\frac{\partial f}{ \partial x_2}(0,1)$?

A1: $0$

A2: $1$

A3: $2$

A4: $3$

Q3: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $$f(x_1, x_2) = x_1^2 \cdot x_2$$ What is $\frac{\partial f}{ \partial x_1}(\tilde{x})$?

A1: $2 \tilde{x}_1 \tilde{x}_2$

A2: $2 x_1 \tilde{x}_2$

A3: $2 x_1 x_2$

A4: $2 \tilde{x}_1$

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