Differentiate the following functions with respect to x :

$\frac{x^{2} + 1}{x + 1}$

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Solution

Using quotient rule, we have

$\frac{d}{d x} (\frac{x^{2} + 1}{x + 1})$

$= \frac{(x + 1) \frac{d}{d x} (x^{2} + 1) - (x^{2} + 1) \frac{d}{d x} (x + 1)}{(x + 1)^{2}}$

$= \frac{(x + 1) \times 2 x - (x^{2} + 1) \times 1}{(x + 1)^{2}}$

$= \frac{2 x^{2} + 2 x - x^{2} - 1}{(x + 1)^{2}}$

$= \frac{x^{2} + 2 x - 1}{(x + 1)^{2}}$

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