Find the difference quotient [tex]\frac{f(x+h)-f(x)}{h}[/tex], where [tex]h \neq 0[/tex], for the function below.

[tex]f(x) = 3x^2 - 3[/tex]

Simplify your answer as much as possible.

[tex]\frac{f(x+h)-f(x)}{h} = \boxed{\phantom{answer}}[/tex]



Answer :

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]