What is meant by $\frac{d ^2y}{dx^2}$?

Question

I have this homework question:

Find $\frac{d^2y}{dx^2}$ for $y = (x^3−5)(2x+3)$.

But I do not know why there are squares in $\frac{d^2y}{dx^2}$, so I cannot solve it. What is meant by this?

Answer 1

The operator $\frac{\mathrm d^2}{\mathrm dx^2}$ is the square of the operator $\frac{\mathrm d}{\mathrm dx}$ and as the latter is "take the derivative", the former means "take the derivative twice in a row" or "take the second derivative".

Answer 2

It is abuse of notation meaning $$ \frac{d\frac{dy}{dx}}{dx} $$where pretending that $d$ and $dx$ are really symbols, you get the "fraction" $\frac{d^2y}{dx^2}$. So it means "differentiate $y$ with respect to $x$, then differentiate the result with respect to $x$", or for short "calculate $y''(x)$".

Answer 3

The notation $\frac{d^2y}{dx^2}$ simply means the second derivative of $y$ with respect to $x$. There are all sorts of notations for differentiation. It would be smart to familiarize yourself with them.

For your problem specifically, you have $$ y=(3x-5)(2x+3) = 6x^2-x-15. $$ Thus, you have $$ y' = 12x-1 $$ and then $$ y'' = 12 $$ for your answer, where the "prime" notation is here used to indicate how many derivatives you are taking.