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Title: Properties for Derivatives
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Series: Real Analysis
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Chapter: Differentiable Functions
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YouTube-Title: Real Analysis 35 | Properties for Derivatives
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Ad-free video: Watch Vimeo video
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Exercise Download PDF sheets
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra35_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: What is not a correct definition for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ being differentiable at $x_0$?
A1: $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{x - x_0}$ exists.
A2: There is a function $\Delta_{f,x_0} : \mathbb{R} \rightarrow \mathbb{R}$ that is continuous at $x_0$ and satifies $f(x) = f(x_0) + (x - x_0) \cdot \Delta_{f,x_0}(x)$ for all $x \in \mathbb{R}$.
A3: There is a number $b \in \mathbb{R}$ and a function $r : \mathbb{R} \rightarrow \mathbb{R}$ that is continuous at $x_0$ with $r(x_0) = 0$ and satifies $f(x) = f(x_0) + (x - x_0) \cdot b + (x - x_0) \cdot r(x)$ for all $x \in \mathbb{R}$.
A4: $f$ is continuous at $x_0$ and the limit $\displaystyle \lim_{x \rightarrow x_0} \frac{f(x-x_0) - f(x_0)}{x - x_0}$ exists.
Q2: Which of the following implications for a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is correct?
A1: $f$ continuous $\Rightarrow$ $f$ differentiable
A2: $f$ differentiable at $x_0$ $\Rightarrow$ $f$ continuous
A3: $f$ differentiable at $x_0$ $\Rightarrow$ $f$ continuous at $x_0$
A4: $f$ continuous at $x_0$ $\Rightarrow$ $f$ differentiable at $x_0$
A5: $f$ continuous at $x_0$ $\Rightarrow$ $f$ differentiable everywhere
Q3: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable at $x_0$. What is the correct product rule?
A1: $(f \cdot g)^\prime(x_0) = f^\prime(x_0) \cdot g^\prime(x_0)$
A2: $(f \cdot g)^\prime(x_0) = f^\prime(x_0) \cdot g^\prime(x_0) + f(x_0) \cdot g(x_0) $
A3: $(f \cdot g)^\prime(x_0) = f^\prime(x_0) \cdot g(x_0) - f(x_0) \cdot g^\prime(x_0) $
A4: $(f \cdot g)^\prime(x_0) = f(x_0) \cdot g^\prime(x_0) + f^\prime(x_0) \cdot g(x_0) $
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Last update: 2025-01
