Question

1. consider the snippet of computation below.

lim_{h
ightarrow0}left\frac{f(x + h)+g(x + h) - f(x)+g(x)}{h}
ight
=lim_{h
ightarrow0}left\frac{f(x + h)-f(x)}{h}+\frac{g(x + h)-g(x)}{h}
ight
=lim_{h
ightarrow0}left\frac{f(x + h)-f(x)}{h}
ight+lim_{h
ightarrow0}left\frac{g(x + h)-g(x)}{h}
ight
which property of differentiation does this help prove?
\frac{d}{dx}f(x)cdot g(x)=f(x)cdot g(x)+g(x)cdot f(x)
\frac{d}{dx}left\frac{f(x)}{g(x)}
ight=\frac{g(x)cdot f(x)-f(x)cdot g(x)}{g^{2}(x)}
\frac{d}{dx}acdot f(x)=acdot f(x)
\frac{d}{dx}f(x)+g(x)=f(x)+g(x)

Explanation:

The given limit - based computation shows the splitting of the limit of the difference - quotient of the sum of two functions $f(x)$ and $g(x)$ into the sum of the limit of the difference - quotients of $f(x)$ and $g(x)$ separately. This is the basis for the sum rule of differentiation. The sum rule states that the derivative of the sum of two functions is the sum of their derivatives.

Answer:

$\frac{d}{dx}[f(x)+g(x)] = f'(x)+g'(x)$