$\text { Use the formula } f ^ { \prime } ( x ) = \lim _ { z \rightarrow x } \frac { f ( z ) - f ( x ) } { z - x } \text { to find the derivative of the function. }$
-$$f ( x ) = \frac { 5 } { x + 6 }$$
A) $\frac { 5 } { ( x + 6 ) ^ { 2 } }$
B) $- \frac { 5 } { ( x + 6 ) ^ { 2 } }$
C) $- \frac { 5 } { ( x + 6 ) }$
D) $- \frac { 5 } { x ^ { 2 } }$

Question

Quizplus · Accepted Answer

The derivative of $f(x) = \frac{5}{x + 6}$ using the definition of the derivative is $- \frac { 5 } { ( x + 6 ) ^ { 2 } }$. This is because when you apply the limit definition and simplify, the negative sign comes from the power rule applied to the denominator, indicating the rate of change decreases as $x$ increases.

- f(x)=5x+6f ( x ) = \frac { 5 } { x + 6 }f(x)=x+65​

- $f ( x ) = \frac { 5 } { x + 6 }$