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    RichardCumbicos
    The document derives the derivative of the function f(x) = √(x + √x) using the definition of the derivative. It takes multiple steps of algebraic manipulation and limiting processes as the change in x, Δx, approaches 0. In the end, the derivative is simplified to an expression containing √x terms divided by other √x terms.

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    The document derives the derivative of the function f(x) = √(x + √x) using the definition of the derivative. It takes multiple steps of algebraic manipulation and limiting processes as the change in x, Δx, approaches 0. In the end, the derivative is simplified to an expression containing √x terms divided by other √x terms.

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    The document derives the derivative of the function f(x) = √(x + √x) using the definition of the derivative. It takes multiple steps of algebraic manipulation and limiting processes as the change in x, Δx, approaches 0. In the end, the derivative is simplified to an expression containing √x terms divided by other √x terms.

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    © All Rights Reserved
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    7 views1 page

    Textbf

    Uploaded by

    RichardCumbicos
    The document derives the derivative of the function f(x) = √(x + √x) using the definition of the derivative. It takes multiple steps of algebraic manipulation and limiting processes as the change in x, Δx, approaches 0. In the end, the derivative is simplified to an expression containing √x terms divided by other √x terms.

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    \textbf{3)Derivar por definicin}\\

    \begin{tabular}{c c c}
    $ \frac{d}{dx} \left( \sqrt{x+ \sqrt{x}}\right) $ & $=$ &
    $\displaystyle \lim_{\bigtriangleup x \to 0}\frac{\sqrt{x+
    \bigtriangleup x + \sqrt{x+ \bigtriangleup x}}-
    \sqrt{x+\sqrt{x}}\rightarrow a}{\bigtriangleup x}$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \lim_{\bigtriangleup x \to 0}
    \frac{\left(\sqrt{x+ \bigtriangleup x + \sqrt{x+\bigtriangleup x}} -
    \sqrt{x+\sqrt{x}} \right)\overline{a}}{\bigtriangleup x
    \overline{a}}$\\
    \vspace{2mm}
    & $=$ & $ \displaystyle \lim_{\bigtriangleup x \to 0}\frac{x+
    \bigtriangleup x + \sqrt{x+ \bigtriangleup x}- x -
    \sqrt{x}}{\overline{a} \bigtriangleup x}$\\
    \vspace{2mm}
    &$=$&$ \displaystyle \lim_{\bigtriangleup x \to 0}\frac{\left(
    \bigtriangleup x + \sqrt{x+\bigtriangleup x} \right)-
    \sqrt{x}\rightarrow \overline{b}}{\overline{a} \overline{b}
    \bigtriangleup x}$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \lim_{\bigtriangleup x \to 0}\frac{[\left(
    \bigtriangleup x + \sqrt{x + \bigtriangleup x} \right)-\sqrt{x} ]\cdot
    \overline{b} }{\overline{a} \overline{b} \bigtriangleup x}$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \lim_{\bigtriangleup x \to 0}
    \frac{\bigtriangleup x^2 + 2 \bigtriangleup x \sqrt{x+\bigtriangleup
    x}+x+ \bigtriangleup x-x}{\overline{a} \overline{b} \bigtriangleup
    x}$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \lim_{\bigtriangleup x \to
    0}\frac{\bigtriangleup x \left( \bigtriangleup x + 2x
    \sqrt{x+\bigtriangleup x}+1 \right) }{\overline{a} \overline{b}
    \bigtriangleup x}$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \lim_{\bigtriangleup x \to 0}
    \frac{\bigtriangleup x + 2x \sqrt{x + \bigtriangleup
    x}+1}{\left(\sqrt{x + \bigtriangleup x + \sqrt{x + \bigtriangleup
    x}}+\sqrt{x+\sqrt{x}} \right) \left( \bigtriangleup x + \sqrt{x+
    \bigtriangleup x} + \sqrt{x} \right) }$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \frac{2x \sqrt{x} + 1}{2 \sqrt{x + \sqrt{x}}
    \cdot 2 \sqrt{x}}$\\
    \vspace{2mm}
    & $=$ & $\displaystyle \frac{2x \sqrt{x}+1}{4 \sqrt{x +
    \sqrt{x}}\cdot \sqrt{x}}$\\
    \end{tabular}
    \\

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