D (x) = (x + 1) (x + 2) ^ 2 (x +3) ^ 3 (x + 4) ^ 4 (x + 5)
Find the derivative at -1
Use log differentiation
$\displaystyle ln[D(x)] = ln[(x+1)(x+2)^2(x+3)^3(x+4)^4(x+5)]$ $\displaystyle = ln(x+1) + 2ln(x+2) + 3ln(x+3) + 4ln(x+4) + ln(x+5)$
Recall that due the chain rule: $\displaystyle \frac{d}{dx}ln[f(x)] = \frac{f'(x)}{f(x)}$
Actually, looking at it I don't think it is differentiable at x=-1 because of the 1/(x+1) term
$\displaystyle D(x) = f(x) \cdot g(x) \cdot h(x) \cdot p(x) \cdot q(x)
$
$\displaystyle D'(x) = f'(x) \cdot g(x) \cdot h(x) \cdot p(x) \cdot q(x)$ $\displaystyle + f(x) \cdot g'(x) \cdot h(x) \cdot p(x) \cdot q(x)$ $\displaystyle + f(x) \cdot g(x) \cdot h'(x) \cdot p(x) \cdot q(x)$ $\displaystyle + f(x) \cdot g(x) \cdot h(x) \cdot p'(x) \cdot q(x)$ $\displaystyle + f(x) \cdot g(x) \cdot h(x) \cdot p(x) \cdot q'(x)$
since $\displaystyle f(x) = (x+1)$ , and $\displaystyle f(-1) = 0$ ...
$\displaystyle D'(-1) = (1)(1^2)(2^3)(3^4)(4)$
You could go right to the definition
$\displaystyle
D'(-1) = \lim_{x \to -1} \frac{D(x) - D(-1)}{x+1} = \lim_{x \to -1} \frac{(x+1)(x+2)^2(x+3)^3(x+4)^4(x+5)}{x+1}
$
$\displaystyle = \lim_{x \to -1} (x+2)^2(x+3)^3(x+4)^4(x+5) = 1^2 \cdot 2^3 \cdot 3^4 \cdot 4$ as Skeeter said.
after differentiating both sides and multiplying both sides by $\displaystyle (x+1)(x+2)^2(x+3)^3(x+4)^4(x+5)$, you'll get y'= $\displaystyle [\frac{1}{x+1} + \frac{2}{x+2}+\frac{3}{x+3}+\frac{4}{x+4}+\frac{1} {x+5}]$ $\displaystyle [(x+1)(x+2)^2(x+3)^3(x+4)^4(x+5)]$
then distribute and plug in x = -1 and you'll see that all the terms will cancel out and become zero except for the first term, since the (x+1)s cancel out. when you plug in x = -1 to what you have left, you'll get $\displaystyle (1^2)(2^3)(3^4)(4)$