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SOVA2 [1]
3 years ago
5

F(x) =

Mathematics
1 answer:
sertanlavr [38]3 years ago
6 0

To make things easier, first rewrite

<em>f(x)</em> = <em>x</em> / (<em>x</em> + 7/<em>x</em>) = <em>x</em> ² / (<em>x</em> ² + 7)

i.e. multiply <em>f</em> by <em>x</em>/<em>x</em>, which is valid because <em>x</em> can't be 0 anyway as it's outside the domain of <em>f</em>.

Now, by the quotient and chain rules, we get

<em />

<em>f '(x)</em> = ((<em>x</em> ² + 7) (<em>x</em> ²)' - <em>x</em> ² (<em>x</em> ² + 7)') / (<em>x</em> ² + 7)²

… = (2<em>x</em> (<em>x</em> ² + 7) - <em>x</em> ² (2<em>x</em>)) / (<em>x</em> ² + 7)²

… = (2<em>x</em> ³ + 14<em>x</em> - 2<em>x </em>³) / (<em>x</em> ² + 7)²

… = 14<em>x</em> / (<em>x</em> ² + 7)²

<em>f ''(x)</em> = ((<em>x</em> ² + 7)² (14<em>x</em>)' - (14<em>x</em>) ((<em>x</em> ² + 7)²)') / ((<em>x</em> ² + 7)²)²

… = (14 (<em>x</em> ² + 7)² - (14<em>x</em>) (2 (<em>x</em> ² + 7)² (<em>x</em> ² + 7)')) / (<em>x</em> ² + 7)⁴

… = (14 (<em>x</em> ² + 7)² - (14<em>x</em>) (2 (2<em>x</em>) (<em>x</em> ² + 7)²)) / (<em>x</em> ² + 7)⁴

… = (14 (<em>x</em> ² + 7)² - 4<em>x</em> (14<em>x</em>) (<em>x</em> ² + 7)²) / (<em>x</em> ² + 7)⁴

… = 14 (<em>x</em> ² + 7)² (1 - 4<em>x</em>) / (<em>x</em> ² + 7)⁴

… = 14 (1 - 4<em>x</em>) / (<em>x</em> ² + 7)²

… = (14 - 56<em>x</em>) / (<em>x</em> ² + 7)²

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