How Do You Apply the Chain and Product Rules in Differentiation?

[andrew.c]

Homework Statement

Differentiate...
$$(3x+4)^7 (7x-1)^3$$
and
simplify

Homework Equations

Chain rule and Product rule

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 doesn't match the answer in the marking

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

Any ideas guys?

 

[HallsofIvy]

Homework Statement

Differentiate...
$$(3x+4)^7 (7x-1)^3$$
and
simplify

Homework Equations

Chain rule and Product rule

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 doesn't match the answer in the marking

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

Any ideas guys?

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.

 

[andrew.c]

How could you possibly do this differentiation correctly (which you did) and not be able to multiply polynomials!

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

Thanks