Question

find and simplify the difference quotient \\( \frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \
eq 0 \\) for the given function. \\( f ( x ) = 3 x ^ { 2 } \\) \\( \frac { f ( x + h ) - f ( x ) } { h } = \square \\) (simplify your answer.)

Explanation:

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

Given \( f(x)=3x^{2} \), substitute \( x\) with \( x + h\):
\( f(x + h)=3(x + h)^{2}=3(x^{2}+2xh+h^{2})=3x^{2}+6xh + 3h^{2} \)

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

$$
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$$

Step3: Simplify the numerator and then divide by \( h \)

The \( 3x^{2}\) terms in the numerator cancel out: \( \frac{6xh+3h^{2}}{h} \).
Factor out \( h \) from the numerator: \( \frac{h(6x + 3h)}{h} \).
Since \( h
eq0 \), we can cancel out the \( h \) terms: \( 6x+3h \)

Answer:

\(6x + 3h\)