fiven $$ f(x)=-6 x+7 $$, use the difference quotient in a table to estimate $$ f^{\prime}(2) $$. $$ f^{\prime}(2)= $$

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We are given the function 
$$
f(x)=-6x+7.
$$
The difference quotient for a point $$ x $$ and a small increment $$ h $$ is given by
$$
\frac{f(x+h)-f(x)}{h}.
$$
To estimate $$ f'(2) $$ using the difference quotient, we will set $$ x=2 $$ and create a table for various small values of $$ h $$.

First, calculate $$ f(2) $$:
$$
f(2)=-6(2)+7=-12+7=-5.
$$

Next, we compute $$ f(2+h) $$:
$$
f(2+h)=-6(2+h)+7=-12-6h+7=-5-6h.
$$

Now, the difference quotient is:
$$
\frac{f(2+h)-f(2)}{h}=\frac{(-5-6h)-(-5)}{h}=\frac{-6h}{h}=-6.
$$

Below is a table with different values of $$ h $$:

| $$ h $$ | $$ f(2+h) $$                    | Difference Quotient $$\frac{f(2+h)-f(2)}{h}$$ |
|---------|---------------------------------|-----------------------------------------------|
| 1       | $$ f(3)=-6(3)+7=-18+7=-11 $$     | $$\frac{-11-(-5)}{1}=\frac{-6}{1}=-6$$          |
| 0.5     | $$ f(2.5)=-6(2.5)+7=-15+7=-8 $$   | $$\frac{-8-(-5)}{0.5}=\frac{-3}{0.5}=-6$$        |
| 0.1     | $$ f(2.1)=-6(2.1)+7=-12.6+7=-5.6 $$ | $$\frac{-5.6-(-5)}{0.1}=\frac{-0.6}{0.1}=-6$$     |
| 0.01    | $$ f(2.01)=-6(2.01)+7=-12.06+7=-5.06 $$ | $$\frac{-5.06-(-5)}{0.01}=\frac{-0.06}{0.01}=-6$$  |

As seen, for each value of $$ h $$, the difference quotient is $$-6$$.

Therefore,
$$
f^{\prime}(2)=-6.
$$
$$ f^{\prime}(2) = -6 $$

			Difference Quotient
0.1		-5.6 - (-5) = -0.6
0.01		-5.06 - (-5) = -0.06
0.001		-5.006 - (-5) = -0.006

fiven , use the difference quotient in a table to estimate .

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