Chain Product Quotient 85739A

1. Let's first state the problem: You want to identify whether a given differentiation rule is the chain rule, product rule, or quotient rule. 2. The **product rule** is used when differentiating the product of two functions $f(x)$ and $g(x)$: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$ 3. The **quotient rule** is used when differentiating the quotient of two functions $f(x)$ and $g(x)$: $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$ 4. The **chain rule** is used when differentiating a composite function $h(x) = f(g(x))$: $$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$$ 5. To identify which rule to use, look at the function: - If it is a product of two functions, use the product rule. - If it is a quotient of two functions, use the quotient rule. - If it is a composition of functions, use the chain rule. 6. Example: For $y = (3x^2 + 1)^5$, since this is a composition of functions, use the chain rule. 7. Example: For $y = (x^2 + 1)(x - 3)$, since this is a product, use the product rule. 8. Example: For $y = \frac{x^2 + 1}{x - 3}$, since this is a quotient, use the quotient rule. This explanation helps you decide which differentiation rule applies based on the function's form.