Question

Which rules should I use to find this derivative?

Answer 1

If I want to find the derivative of:
\[f(x)=\frac{ (3x ^{2}+1)(2x-1) }{ (5x+4) }\]
Which rules would I use and in which order?

Answer 2

I know I could just foil out the top, but would it make sense do to the Product Rule as well as the Quotient Rule? If so, which order should I use them in?

Answer 3

to simplify, i'd suggest you foil it then do the quotient rule

Answer 4

What would I do if it were too obnoxious to do the foil rule? For example:
\[f(x)=\frac{ (3x-4)^{9} }{ 5x-2 }\]

Answer 5

with that you can just use the quotient rule since there is (3x-4)^9 instead of something like (3x-2)^8*(2x-3)^7, that would require both product and quotient rule

Answer 6

I guess what I'm asking is how do I go about finding a derivative when I'm required to use both the quotient and product rule.

Answer 7

(Because I guess actually that last example would require the chain rule, not the product rule.)

Answer 8

\[f(x)=\frac{ \color{orangered}{(3x ^{2}+1)(2x-1)} }{ (5x+4) }\]

In your first example, you have the product rule `within` the quotient rule.
So you would apply the quotient rule first, and then whenever you're asked to differentiate this orange part, you apply the product rule.

Answer 9

And yes, beyond that, you would need to apply the chain rule after applying the product rule :)

Answer 10

^

Answer 11

Thank you so much! :)

Answer 12

np.
If you need to see that one worked out, lemme know c: