9.7 Second-Order Differentiation, Turning Points, Maximum and Minimum Points

1. When a function y = x³ + x² – 3x + 6 is differentiated with respect to x, the derivative  $\frac{dy}{dx}=3x^2+2x-3$

2. The second function   $\frac{dy}{dx}$ can be differentiated again with respect to x. This is called the second derivative of y with respect to x and can be written as $\frac{d^{2}y}{d{x}^2}$ .

3. Take note that   $\frac{d^{2}y}{d{x}^2}\ne {\left(\frac{dy}{dx}\right)}^2$ .

If y = 4x³ – 7x² + 5x – 1,

The first derivative   $\frac{dy}{dx}=12x^2-14x+5$

The second derivative    $\frac{d^{2}y}{d{x}^2}=24x-14$