Answer:
[tex]\frac{f(x) - f(a)}{x - a}=\frac{-1}{xa}[/tex]
Step-by-step explanation:
Given
[tex]f(x) = \frac{1}{x}[/tex]
Required
Evaluate [tex]\frac{f(x) - f(a)}{x - a}[/tex]
We have:
[tex]f(x) = \frac{1}{x}[/tex]
Next, we calculate f(a)
Substitute a for x in [tex]f(x) = \frac{1}{x}[/tex]
[tex]f(a) = \frac{1}{a}[/tex]
Substitute values for f(x) and f(a) in [tex]\frac{f(x) - f(a)}{x - a}[/tex]
[tex]\frac{\frac{1}{x} - \frac{1}{a}}{x - a}[/tex]
Take LCM of the numerator
[tex]\frac{\frac{a - x}{xa} }{x - a}[/tex]
Split:
[tex]\frac{a - x}{xa} /{x - a}[/tex]
Convert / to *
[tex]\frac{a - x}{xa} *\frac{1}{x - a}[/tex]
Express a - x as -(x-a)
[tex]\frac{-(x - a)}{xa} *\frac{1}{x - a}[/tex]
Divide numerator and denominator by x - a
[tex]\frac{-1}{xa} *\frac{1}{1}[/tex]
[tex]\frac{-1}{xa}[/tex]
Hence:
[tex]\frac{f(x) - f(a)}{x - a}=\frac{-1}{xa}[/tex]
