Techniques of Differentiation - Exercise 2

Exercise 2. Find the derivative of

\begin{displaymath}f(x) = \frac{\sqrt{x}}{x^2 - x + 3}\end{displaymath}

Answer. We will use the quotient rule to get

\begin{displaymath}f'(x) = \frac{(\sqrt{x})' (x^2 - x+3) - (2x - 1)\sqrt{x}}{(x^2 - x + 3)^2}\cdot\end{displaymath}

Since $(\sqrt{x})' = \displaystyle \frac{1}{2}
\frac{1}{\sqrt{x}}$, we get

\begin{displaymath}(\sqrt{x})' (x^2 - x+3) - (2x - 1)\sqrt{x}= -\frac{3}{2} x\sqrt{x} + \frac{\sqrt{x}}{2} + \frac{3}{2\sqrt{x}} \cdot\end{displaymath}

Putting things together one will get f '(x).

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