Suppose f and g are differentiable functions. The quotient rule is proved below.
ID: 2859513 • Letter: S
Question
Suppose f and g are differentiable functions. The quotient rule is proved below. Next to each line, a space is provided. In each space, we will list the reason that line is equal to the previous line. Sometimes I have provided the reason. If the space is blank, you get to fill in the reason. If you need more space, just number the blank, you get to fill in the reason. If you need more space, just number the blanks and write your reasons on a separate piece of paper next to the corresponding numbers. The reasons the 6 limits evaluate as they do are provided below (fill in the missing reasons): g(x) is constant with respect to h definition of derivative 1/f(x) f(x + h) is continuous so we can evaluate it by plugging 0 in for hExplanation / Answer
2) By taking Least common divisor for denominators
3) Add and subtract f(x)g(x) so that value equals the same
4) Rewriting by grouping and making denominator one
5) g(x) taken as common factor for first two terms and f(x) for second two terms
6) Writing difference quotient f(x) and g(x) as an explicit factor
7) Distribute the limits to all terms
8) Apply limits using derivatives definition