• Title: Properties for Derivatives

  • Series: Real Analysis

  • Chapter: Differentiable Functions

  • YouTube-Title: Real Analysis 35 | Properties for Derivatives

  • Bright video: https://youtu.be/wp-s9c1IKhI

  • Dark video: https://youtu.be/5nEsNSpPbno

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ra35_sub_eng.srt missing

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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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