Question

Derivative calculus


`f(x) = \frac{x {}^(2) + 10x + 25}{x {}^(2) - 25 }`
Im begging you guys ​

Answer 1

`\mathsf{ \blue{ f'(x) = \frac{ - 10}{ {(x - 5)}^(2) } } }`

Explanation:

`\mathsf{f(x) = \frac{ {x}^(2) + 10x + 25}{ {x}^(2) - 25} }`

the above expression can be reduced to simpler terms

• x² - 25 = (x + 5)(x - 5)
• x² + 10x + 25 = (x + 5)²

`\mathsf{\implies f(x) = \frac{ {(x + 5)}^(2) }{(x + 5)(x - 5)} }`

(x + 5)² can be written as (x + 5)(x + 5)

`\mathsf{\implies f(x) = \frac{ \cancel{(x + 5)}(x + 5)}{\cancel{(x + 5)}(x - 5)} }`

`\mathsf{\implies f(x) = (x + 5)/(x - 5) }`

Derivative of a fraction
`\mathsf{(u)/(v)}` is

• 
  `\boxed { \red {\mathsf{ (v((du)/(dx)) \: - \: u((dv)/(dx)))/(v^2)} }}`

`\mathsf{\implies f'(x) = \frac{(x - 5) - (x + 5)}{(x - 5) {}^(2) } }`

`\mathsf{\implies f '(x) = \frac{x - 5 - x - 5}{ {(x - 5)}^(2) } }`

`\mathsf{\implies f'(x) = \frac{ - 10}{ {(x - 5)}^(2) } }`