# Differentiate and simplify!

1. Apr 3, 2009

### andrew.c

1. The problem statement, all variables and given/known data
Differentiate...
$$(3x+4)^7 (7x-1)^3$$
and
simplify

2. Relevant equations

Chain rule and Product rule

3. The attempt at a solution

I got (splitting the components up to substitute into the product rule) and using the chain rule

\begin{align*} \\f(x) = (3x+4)^7\\ f'(x) = 21(3x+4)^6\\ g(x) = (7x-1)^3\\ g'(x) = 21(7x-1)^2 \end{align*}

and so, using the product rule...
\begin{align*} \\f'(x)g(x) + f(x)g'(x)\\ =21(3x+4)^6 (7x-1)^3 + 21(3x+4)^7 (7x-1)^2\\ =(3x+4)^6 (7x-1)^3 + (3x+4)^7 (7x-1)^2\\ \end{align*}

and now I don't know how to simplify further.
I got it down to 10x+3, but this doesnt match the answer in the marking

$$21(3x+4)^6 (7x-1)^2 (10x+3)$$

Any ideas guys?

2. Apr 3, 2009

### HallsofIvy

Staff Emeritus
How could you possibly do this differentiation correctly (which you did) and not be able to multiply polynomials! You product, after multiplying out, will involve x9. It certainly is not "10x+ 3"!

Notice that your $(3x+4)^6(7x-1)^3+ (3x+4)^7(7x-1)^2$ has at least 6 factors of 3x-4 and 2 factors of 7x-1 in each term. Take them out and you have left exactly that "10x+3" you mentioned.

3. Apr 3, 2009

### andrew.c

Yeah, just had a look through this again and that was a really stupid mistake! I guess thats what hours of maths can do to you!

Thanks