${d^2\over dx^2}$ notation problem

Question

How would I figure out the following problem.

Find

$\frac{d^2}{dx^2}$ $[(x^3-1)\frac{d^2}{dx^2}(6x-x^5)$

This is what I did

I took the derivative of $(6x-x^5)$ and got $(-20x^3)$

I did $(x^3-1)(-20x^3)$ and got $(-20x^6+20x^3)$

Then I took the derivative of that twice and got

$(-600x^4+120x)$

But did I do this correctly?

Answer 1

Assuming you mean to close the left bracket on the right of the entire expression:$$\frac{d^2}{dx^2}\left[(x^3-1)\frac{d^2}{dx^2}(6x-x^5)\right]$$

Then your result is correct, but I suspect you meant that you took the derivative twice (sequentially), of $(6x - x^5)$, getting $$d/dx (6 - 5x^4) = -20x^3.$$

Then you correctly multiplied $(x^3 - 1)(-20x^3)$.

And then took the derivative of that product, twice, to get your answer: $-600x^4 + 120x$

Nice job.

Answer 2

Take the derivative of the derivative of $6x-x^5$, then multiply the result by $x^3-1$, and then take the derivative of the derivative of that.

You should get $120x-600x^4$.