Given the function $f(x) = 1 - x^2$, we want to find the difference quotient from $x$ to $h + h$ (which is essentially $x + 2h$).

The difference quotient is given by:

$\frac{f(x+2h) - f(x)}{(x+2h) - x}$

Step 1: Compute $f(x+2h)$

Substitute $x + 2h$ into $f(x)$:

$f(x + 2h) = 1 - (x + 2h)^2$

Expanding $(x + 2h)^2$:

$(x + 2h)^2 = x^2 + 4xh + 4h^2$

So,

$f(x + 2h) = 1 - (x^2 + 4xh + 4h^2) = 1 - x^2 - 4xh - 4h^2$

Step 2: Compute the difference $f(x+2h) - f(x)$

Now subtract $f(x)$ from $f(x+2h)$:

$f(x + 2h) - f(x) = (1 - x^2 - 4xh - 4h^2) - (1 - x^2)$

Simplifying:

$f(x + 2h) - f(x) = 1 - x^2 - 4xh - 4h^2 - 1 + x^2$

$f(x + 2h) - f(x) = -4xh - 4h^2$

Step 3: Divide by $(x + 2h) - x = 2h$

Now, compute the difference quotient:

$\frac{f(x+2h) - f(x)}{2h} = \frac{-4xh - 4h^2}{2h}$

Simplifying:

$\frac{-4xh - 4h^2}{2h} = -2x - 2h$

Final Answer:

The difference quotient from $x$ to $x + 2h$ is:

$-2x - 2h$

Would you like more details or have any questions?

Here are 5 related questions you can explore:

1. What is the general form of the difference quotient for any function $f(x)$?
2. How do you apply the difference quotient to polynomial functions?
3. What happens to the difference quotient as $h \to 0$?
4. How is the difference quotient related to the derivative of a function?
5. Can you find the derivative of $f(x) = 1 - x^2$ using the difference quotient?

Tip: Always simplify algebraic expressions carefully before substituting them into the difference quotient to avoid mistakes.