# How do you know whether to use chain, product or quotient rule?

Mar 29, 2015

Think about the order of operations. What's the last operation you would do if you had a number and were doing arithmetic.

$f \left(x\right) = 5 {x}^{2} \sin x$
Last operation is multiply. Use the product rule.

$g \left(x\right) = \frac{4 {x}^{2} + 3}{2 x - 1}$
Last operation is division, use the quotient rule.

$h \left(x\right) = {\left(5 {x}^{2} + 1\right)}^{9}$
Last operation is raise to the 9th power, but it's not just ${x}^{9}$, use the chain rule.

$F \left(x\right) = 3 {x}^{5} \cos \left(7 x - 1\right)$ Last operation is multiply, use the product rule, but, whe you take the derivative of $\cos \left(7 x - 1\right)$, you'll need the chain rule.

$G \left(x\right) = {\left(\frac{3 x + 2}{x - 5}\right)}^{7}$

Use the derivative of the 7th power and (because it's not simply ${x}^{7}$, you'll also need the chain rule and to find the derivative of $\frac{3 x + 2}{x - 5}$ (the "inside function")

If you rewrite $G \left(x\right) = {\left(3 x + 2\right)}^{7} / {\left(x - 5\right)}^{7}$, then the last operation is division, so we'll need the quotient rule. (And the power and chain rules).