$$\text { Find and simplify the difference quotient } \frac { f ( a + h ) - f ( a ) } { h } , h \neq 0 \text { for the given function. }$$
-$f ( x ) = \frac { 1 } { 3 x }$
A) $\frac { - 1 } { a ( a + h ) }$
B) 0
C) $\frac { - 1 } { 3 a ( a + h ) }$
D) $\frac { 1 } { 3 a }$

Question

Quizplus · Accepted Answer

To find the difference quotient for $f(x) = \frac{1}{3x}$, we substitute $f(a+h)$ and $f(a)$ into the formula: $$\frac{f(a+h) - f(a)}{h} = \frac{\frac{1}{3(a+h)} - \frac{1}{3a}}{h}$$ Simplifying this expression, we get: $$\frac{\frac{a - (a+h)}{3a(a+h)}}{h} = \frac{-h}{3ah(a+h)} = \frac{-1}{3a(a+h)}$$ This matches choice C.