Question

Find the derivative for the function.

Answer 1

The rule for deriving a multiplication is

$(f(x)\cdot g(x))' = f'(x)g(x)+f(x)g'(x)$

Also, we can forget about the factor 4 for now, since we have

$(4f(x)\cdot g(x))' = 4(f'(x)g(x)+f(x)g'(x))$

So, we will just multiply everything by 4 at the end. Our functions are

$f(x) = (x^7-9)^{12},\quad g(x) = (4x+8)^{11}$

For both derivatives we will use the rule

$(f(x)^n)' = n\cdot f(x)^{n-1}\cdot f'(x)$

So, we have

$f'(x) = 12\cdot(x^7-9)^{11}\cdot 7x^6,\quad g'(x) = 11\cdot(4x+8)^{10}\cdot 4$

We can simplify those expression a little bit:

$f'(x) = 84x^6(x^7-9)^{11},\quad g'(x) = 44(4x+8)^{10}$

The formula $f'(x)g(x)+f(x)g'(x)$ thus becomes

$84x^6(x^7-9)^{11}\cdot (4x+8)^{11} + (x^7-9)^{12} \cdot 44(4x+8)^{10}$

And so the final answer is

$4(84x^6(x^7-9)^{11}\cdot (4x+8)^{11} + (x^7-9)^{12} \cdot 44(4x+8)^{10})$

If you simplify this expression by factoring common terms, you will see that the correct answer is the first one.