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

[tex]\[ f(x) = -4x^2 - 2x + 6 \][/tex]

Simplify your answer as much as possible.

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



Answer :

To solve for the difference quotient [tex]\(\frac{f(x+h)-f(x)}{h}\)[/tex] where [tex]\(h \neq 0\)[/tex] for the function [tex]\( f(x) = -4x^2 - 2x + 6 \)[/tex], follow these steps:

1. Substitute [tex]\(x + h\)[/tex] into the function [tex]\(f(x)\)[/tex]:

Given the function [tex]\(f(x) = -4x^2 - 2x + 6\)[/tex], calculate [tex]\(f(x + h)\)[/tex]:
[tex]\[ f(x + h) = -4(x + h)^2 - 2(x + h) + 6 \][/tex]

2. Expand [tex]\((x + h)^2\)[/tex]:

[tex]\((x + h)^2 = x^2 + 2xh + h^2\)[/tex].

Substitute into [tex]\(f(x + h)\)[/tex]:
[tex]\[ f(x + h) = -4(x^2 + 2xh + h^2) - 2(x + h) + 6 \][/tex]

3. Distribute and simplify:

Distribute the [tex]\(-4\)[/tex] and [tex]\(-2\)[/tex] in the expression:
[tex]\[ f(x + h) = -4x^2 - 8xh - 4h^2 - 2x - 2h + 6 \][/tex]

4. Compute the difference [tex]\(f(x + h) - f(x)\)[/tex]:

Subtract [tex]\(f(x) = -4x^2 - 2x + 6\)[/tex] from [tex]\(f(x + h)\)[/tex]:
[tex]\[ f(x + h) - f(x) = (-4x^2 - 8xh - 4h^2 - 2x - 2h + 6) - (-4x^2 - 2x + 6) \][/tex]

Simplify the expression by combining like terms:
[tex]\[ f(x + h) - f(x) = -4x^2 - 8xh - 4h^2 - 2x - 2h + 6 + 4x^2 + 2x - 6 \][/tex]

After canceling terms, this simplifies to:
[tex]\[ f(x + h) - f(x) = -8xh - 4h^2 - 2h \][/tex]

5. Form the difference quotient:

The difference quotient is given by:
[tex]\[ \frac{f(x + h) - f(x)}{h} = \frac{-8xh - 4h^2 - 2h}{h} \][/tex]

6. Simplify the quotient:

Factor out [tex]\(h\)[/tex] from the numerator:
[tex]\[ \frac{-8xh - 4h^2 - 2h}{h} = \frac{h(-8x - 4h - 2)}{h} \][/tex]

Cancel [tex]\(h\)[/tex] from the numerator and the denominator:
[tex]\[ -8x - 4h - 2 \][/tex]

Therefore, the simplified difference quotient is:
[tex]\[ \frac{f(x+h)-f(x)}{h} = -8x - 4h - 2 \][/tex]