To find the difference quotient for the function [tex]\( f(x) = 3x^2 - 3 \)[/tex], we need to follow these steps:
1. Evaluate [tex]\( f(x + h) \)[/tex]:
Substitute [tex]\( x + h \)[/tex] into the function [tex]\( f(x) \)[/tex].
[tex]\[
f(x + h) = 3(x + h)^2 - 3
\][/tex]
Expand [tex]\( (x + h)^2 \)[/tex]:
[tex]\[
(x + h)^2 = x^2 + 2xh + h^2
\][/tex]
Therefore,
[tex]\[
f(x + h) = 3(x^2 + 2xh + h^2) - 3 = 3x^2 + 6xh + 3h^2 - 3
\][/tex]
2. Calculate [tex]\( f(x + h) - f(x) \)[/tex]:
Substitute the expressions for [tex]\( f(x + h) \)[/tex] and [tex]\( f(x) \)[/tex]:
[tex]\[
f(x + h) - f(x) = [3x^2 + 6xh + 3h^2 - 3] - [3x^2 - 3]
\][/tex]
Cancel out the like terms [tex]\( 3x^2 \)[/tex] and [tex]\(-3\)[/tex]:
[tex]\[
f(x + h) - f(x) = 3x^2 + 6xh + 3h^2 - 3 - 3x^2 + 3
\][/tex]
Simplify the expression:
[tex]\[
f(x + h) - f(x) = 6xh + 3h^2
\][/tex]
3. Form the difference quotient [tex]\( \frac{f(x + h) - f(x)}{h} \)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2}{h}
\][/tex]
Factor out [tex]\( h \)[/tex] in the numerator:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{h(6x + 3h)}{h}
\][/tex]
4. Simplify the difference quotient:
Since [tex]\( h \neq 0 \)[/tex], we can cancel [tex]\( h \)[/tex] in the numerator and denominator:
[tex]\[
\frac{f(x + h) - f(x)}{h} = 6x + 3h
\][/tex]
Thus, the simplified difference quotient is:
[tex]\[
\frac{f(x + h) - f(x)}{h} = 6x + 3h
\][/tex]
