Question

Need help with an AP-Calculus question

Answer 1

We could use the formula for the derivative of a product of two functions, or simply do the multiplication and find the derivative of each term, but that would be too laborious.

We know that
`(x^n)'=nx^(n-1)`. Therefore if a term in the second derivative of a function is squared, then in the orignal function that term must be raised to the 4th power.

In the given function we can get terms of the 4th degree by multiplying
`x^3\cdot(-8x)` or
`2x\cdot 3x^3`.

`x^3\cdot(-8x)=-8x^4\\2x\cdot 3x^3=6x^4`

Now we add them

`-8x^4+6x^4=-2x^4`

And now we need to find the second derivative of this term only.

`(-2x^4)''=(-8x^3)'=-24x^2`

So, the coefficient is -24.