1. consider the snippet of computation below.\n lim_{h\rightarrow0}left\frac{f(x + h)+g(x + h)…

1. consider the snippet of computation below.\n lim_{h\rightarrow0}left\frac{f(x + h)+g(x + h) - f(x)+g(x)}{h}\right\n=lim_{h\rightarrow0}left\frac{f(x + h)-f(x)}{h}+\frac{g(x + h)-g(x)}{h}\right\n=lim_{h\rightarrow0}left\frac{f(x + h)-f(x)}{h}\right+lim_{h\rightarrow0}left\frac{g(x + h)-g(x)}{h}\right\nwhich property of differentiation does this help prove?\n \frac{d}{dx}f(x)cdot g(x)=f(x)cdot g(x)+g(x)cdot f(x)\n \frac{d}{dx}left\frac{f(x)}{g(x)}\right=\frac{g(x)cdot f(x)-f(x)cdot g(x)}{g^{2}(x)}\n \frac{d}{dx}acdot f(x)=acdot f(x)\n \frac{d}{dx}f(x)+g(x)=f(x)+g(x)

1. consider the snippet of computation below.\n lim_{h\rightarrow0}left\frac{f(x + h)+g(x + h) - f(x)+g(x)}{h}\right\n=lim_{h\rightarrow0}left\frac{f(x + h)-f(x)}{h}+\frac{g(x + h)-g(x)}{h}\right\n=lim_{h\rightarrow0}left\frac{f(x + h)-f(x)}{h}\right+lim_{h\rightarrow0}left\frac{g(x + h)-g(x)}{h}\right\nwhich property of differentiation does this help prove?\n \frac{d}{dx}f(x)cdot g(x)=f(x)cdot g(x)+g(x)cdot f(x)\n \frac{d}{dx}left\frac{f(x)}{g(x)}\right=\frac{g(x)cdot f(x)-f(x)cdot g(x)}{g^{2}(x)}\n \frac{d}{dx}acdot f(x)=acdot f(x)\n \frac{d}{dx}f(x)+g(x)=f(x)+g(x)

Answer

Brief Explanations:

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:

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