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.