Question

use the quotient rule with Generalized power rule to find the exact value of f'(2) for the function
f'(x)=(4x+3)^3/(2x-1)

Answer 1

just because I am curious, f(x) is a function, or is it already a derivative? Wether or not, it will change nothing about the answer.

\[ \large f'(x)= \frac{3(4x+3)^2\cdot 4(2x-1) -2\cdot (4x+3)^3}{(2x-1)^2} \]

Answer 2

Just here again for reminding, the quotient rule is:
\[ \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2} \]

Answer 3

ok

Answer 4

but I thought (2x-1) suspost to came first on top.

Answer 5

\[ u= (4x+3)^3 \rightarrow \frac{du}{dx}=u'=3(4x+3)^2 \cdot 4 = 12(4x+3)^2 \\v=(2x-1) \rightarrow v'=2 \]

Answer 6

the order of multiplication doesn't matter if that's what you mean, just be careful with subtraction

Answer 7

still don't get it what did you get for your anserw?

Answer 8

\[ \frac{12(4x+3)^2(2x-1)-2(4x+3)^3}{(2x-1)^2}\]

Answer 9

Do you know how to apply the chain rule in here?

Answer 10

it just says use quotient rule and generalized power rule to find the exact value.

Answer 11

do u mean factor theorem @hooverst ????

Answer 12

oh the generalized power rule is something like the chain rule. For instance, you sure know how to derivate \[x^2\] and you could also derive the following \(( x+3)^2\) by just multiplying the binomial out, but the generalized power rule works for higher orders two, for instance for \((x^2+4x+3)^{101}\) but we don't have to do such an example here, just make a substitution for the inside and derive it normally, then back substitute and multiply with the derived substitution.

Answer 13

If you know about Leibniz Notation you can see how that works yourself.

Answer 14

Let me show you how to do it anyway, you don't have to do all these steps every time you see such a function though, just be aware about that it is the generalized power rule or for all sorts of functions, as the chain rule

\[y=(4x+3)^3 \ \text{let} \ u=4x+3 \rightarrow \frac{du}{dx}=4
\\y=u^3 \rightarrow \frac{dy}{du}=3u^2
\\
\frac{du}{dx}\cdot \frac{dy}{du}=\frac{dy}{dx}=12u^2 = 12(4x+3)^2\]

Answer 15

ok got it.