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2.15: Tareas- Ejemplos de la Definición del Derivado

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    1. Simplifica cada expresión que implique fracciones o expresiones racionales.
      1. \((x+1) \cdot \left( \frac{1}{3} + \frac{x}{x+1} \right)\)
        \(=\frac{x+1}{3} + x = \frac{4x+1}{3}\)
        ans
      2. \(\cfrac{\cfrac{1}{3} + 1}{1-\cfrac{1}{3}}\)
        \(2\)
        ans
      3. \(\cfrac{x + 1}{\cfrac{1}{x}}\)
        \(x^2 + x\)
        ans
      4. \(\cfrac{\cfrac{1}{x} - \cfrac{1}{x+1}}{\cfrac{1}{x} + \cfrac{1}{x+1}}\)
        \(\frac{1}{2x+1}\)
        ans
      5. \(\cfrac{\cfrac{2}{x} - \cfrac{1}{x}}{\cfrac{1 - y}{y}}\)
        \(\frac{y}{x - xy}\)
        ans
    2. En cada caso, usa la definición de la derivada para encontrar\(f'(x)\) (en otras palabras, ¡toma la derivada!)
      1. \(f(x) = 3x - 5 \)
        \(3\)
        ans
      2. \(f(x) = \frac{1}{2} x + 1 \)
        ans
      3. \(f(x) = 2 x^2\)
        \(4x\)
        ans
      4. \(f(x) = (x^2 + x)\)
        \(2x + 1\)
        ans
      5. \(\frac{d}{dx} (e^x)\)(pista: de la tarea de ayer, tenemos\(\lim_{h \to 0} \frac{e^h - 1}{h} = 1\))
        \(e^x\)
        ans
      6. \(f(x) = x^3\)
        \(x^3\)
        ans
    3. En cada caso, usa la definición de la derivada para encontrar\(f'(x)\) (en otras palabras, ¡toma la derivada!). Cada uno de estos es como uno de los “problemas duros” (haga clic aquí)
      1. \(f(x) = 2x^4\)
        \(8x^3\)
        ans
      2. \(f(x) = \sqrt{2x}\)
        \(\frac{1}{\sqrt{2x}}\)
        ans
      3. \(f(x) = \frac{2}{x} \)
        \(-\frac{2}{x^2}\)
        ans
      4. \(f(x) = \sqrt{x+1}\)
        \(\frac{1}{2 \sqrt{x+1}}\)
        ans
      5. \(f(x) = \frac{1}{x+1} \)
        \(-\frac{1}{(x+1)^2}\)
        ans
      6. \(f(x) = \frac{1}{\sqrt{x}}\)
        \(\frac{-1}{2 x \sqrt{x}}\)
        ans
    4. Recordemos la derivada de\(f(x) = x^2\) está dada por\(2x\).
      1. Demostrar que la derivada de\(g(x) = x^2+1\) está\(2x\) utilizando la definición de la derivada. ¿Se puede encontrar una razón intuitiva por la que\(f(x)\) y\(g(x)\) tendría el mismo derivado?
        Agregar una constante mueve la curva hacia arriba o hacia abajo, pero ese desplazamiento no afecta la pendiente de la línea tangente
        ans
      2. Encuentra otra función cuya derivada sea\(2x\), distinta de\(f(x)\) y\(g(x)\).
        \(x^2 + c\)por cualquier valor\(c\)
        ans

    This page titled 2.15: Tareas- Ejemplos de la Definición del Derivado was last modified on Sun, 30 Oct 2022 16:01:17 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform.