[Math] Quotient vs Product Rule

Question

You are asked to differentiate
$$
y = \frac{x – 1}{x + 1}
$$
Looking at the question, I'm thinking I could solve this question using the product rule by making $\tfrac{1}{x + 1}$ into $(x + 1)^{-1}$. Is there something I am not conceptually understanding about the two rules?

Answer 1

The quotient rule and the product rule are the same thing. In particular, the quotient rule follows from the product rule and the chain rule. Recall that the product rule states that if $h(x) = f(x) g(x)$, then $$h'(x) = f'(x) g(x) + f(x) g'(x).$$ Also recall that the chain rule states that if $h(x) = f(g(x))$, then $$h'(x) = f'(g(x))g'(x).$$ Therefore, if $$h(x) = f(x)/g(x) = f(x)\cdot \frac{1}{g(x)},$$ then the product rule gives $$h'(x) = f'(x) \cdot \frac{1}{g(x)} + f(x) \frac{d}{dx}\left[\frac{1}{g(x)}\right].$$ Then the chain rule applied to the second term gives $$\frac{d}{dx}\left[\frac{1}{g(x)}\right] = -(g(x))^{-2} g'(x).$$ Therefore, $$h'(x) = \frac{f'(x)}{g(x)} - \frac{f(x)g'(x)}{g(x)^2} = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}.$$