Question

find and simplify the difference quotient \\( \frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \
eq 0 \\) for the given function.

\\( f ( x ) = - 9 x \\)

\\( \frac { f ( x + h ) - f ( x ) } { h } = \square \\) (simplify your answer.)

Explanation:

Step1: Find \( f(x + h) \)

Given \( f(x)=-9x \), substitute \( x\) with \( x + h\) in \( f(x)\).
\( f(x + h)=-9(x + h)=-9x-9h \)

Step2: Substitute \( f(x + h)\) and \( f(x)\) into the difference quotient formula

\(\frac{f(x + h)-f(x)}{h}=\frac{(-9x - 9h)-(-9x)}{h}\)
Simplify the numerator: \((-9x-9h + 9x)\)
\(\frac{-9x-9h + 9x}{h}=\frac{-9h}{h}\)

Step3: Simplify the expression

Since \(h
eq0\), cancel out the common factor \(h\) in the numerator and denominator.
\(\frac{-9h}{h}=-9\)

Answer:

\(-9\)