Let F(x) = (x+1) / (x^2-16).

Use the definition of the derivative to compute f'(x) and give its domain. Show all steps.

Show me that you know the formula.

Show me that you can do the substitution and we can go from there

f'(x) = [f(x+h)-f(x)] / h

F'(x) = [(x+h)+1] / (x+h)^2-16]-[(x+1)/X^2-16] /h

F'(x) = [(x+h) +1](x^2)-[(x+h)^2-16](x+1) /h

To compute the derivative of F(x) using the definition of the derivative, we will apply the formula:

f'(x) = lim(h->0) [f(x+h) - f(x)] / h,

where f(x) is the function we want to differentiate.

In this case, f(x) = (x+1) / (x^2-16). Plugging this into the derivative definition, we have:

f'(x) = lim(h->0) [f(x+h) - f(x)] / h
= lim(h->0) [{(x+h+1) / ((x+h)^2-16)} - {(x+1) / (x^2-16)}] / h.

Now, let's simplify this expression step by step.

1. Find f(x+h):
f(x+h) = (x+h+1) / ((x+h)^2-16).

2. Plug f(x+h) and f(x) back into the derivative definition:
f'(x) = lim(h->0) [(x+h+1) / ((x+h)^2-16) - (x+1) / (x^2-16)] / h.

3. Apply the least common denominator to the expression:
f'(x) = lim(h->0) [(x+h+1)(x^2-16) - (x+1)((x+h)^2-16)] / h(x^2-16).

4. Expand and simplify the numerator:
f'(x) = lim(h->0) [x(x^2-16) + h(x^2-16) + (x^2-16) - (x+1)((x+h)^2-16)] / h(x^2-16)
= lim(h->0) [x^3 - 16x + hx^2 - 16h + x^2 - 16 - x^3 - 2x^2h - h^2 - 16 + 16x + xh^2 - 16h] / h(x^2-16)
= lim(h->0) [-2x^2h - h^2] / (x^2-16)
= lim(h->0) [h(-2x^2 - h)] / (x^2-16)
= lim(h->0) [-2x^2h - h^2] / (x^2-16).

5. Factor out h:
f'(x) = lim(h->0) h(-2x^2 - h) / (x^2-16)
= lim(h->0) -2x^2 - h.

6. Take the limit as h approaches 0:
f'(x) = -2x^2.

So, the derivative of F(x) is f'(x) = -2x^2.

Lastly, let's consider the domain of f'(x). Since f'(x) is a polynomial, it is defined for all real numbers. Therefore, the domain of f'(x) is (-∞, ∞).