Higher-Order Derivatives

Differentiating a function once gives its first derivative, $\dfrac{dy}{dx}$ or $f'(x)$. Differentiating that result again, with respect to $x$ once more, gives the second derivative, written $\dfrac{d^2y}{dx^2}$, $f''(x)$, or $y''$. Third, fourth and further derivatives are formed the same way, by repeated differentiation, though this chapter's business applications need only up to the second derivative.

Example: for $y = 2x^5 - 3x^3 + 4x$:

$\frac{dy}{dx} = 10x^4 - 9x^2 + 4, \qquad \frac{d^2y}{dx^2} = \frac{d}{dx}\big(10x^4-9x^2+4\big) = 40x^3 - 18x$ …