Answer:
[tex] \mathsf{ \blue{ f'(x) = \frac{ - 10}{ {(x - 5)}^{2} } } }[/tex]
Step-by-step explanation:
[tex] \mathsf{f(x) = \frac{ {x}^{2} + 10x + 25}{ {x}^{2} - 25} } [/tex]
the above expression can be reduced to simpler terms
- x² - 25 = (x + 5)(x - 5)
- x² + 10x + 25 = (x + 5)²
[tex]\mathsf{\implies f(x) = \frac{ {(x + 5)}^{2} }{(x + 5)(x - 5)} } [/tex]
(x + 5)² can be written as (x + 5)(x + 5)
[tex]\mathsf{\implies f(x) = \frac{ \cancel{(x + 5)}(x + 5)}{\cancel{(x + 5)}(x - 5)} } [/tex]
[tex]\mathsf{\implies f(x) = \frac{x + 5}{x - 5} }[/tex]
Derivative of a fraction [tex] \mathsf{\frac{u}{v}} [/tex] is
- [tex]\boxed { \red {\mathsf{ \frac{v(\frac{du}{dx}) \: - \: u(\frac{dv}{dx})}{v^2}} }}[/tex]
[tex]\mathsf{\implies f'(x) = \frac{(x - 5) - (x + 5)}{(x - 5) {}^{2} } }[/tex]
[tex] \mathsf{\implies f '(x) = \frac{x - 5 - x - 5}{ {(x - 5)}^{2} } }[/tex]
[tex] \mathsf{\implies f'(x) = \frac{ - 10}{ {(x - 5)}^{2} } } [/tex]
