Example 1 : If y = $x^{3} -3x^{2} + 8x $ then find $\frac{\text{d}^{2}y}{\text{d}x^{2}}$
Solution : y = $x^{3} -3x^{2} + 8x $
First we will find $\frac{\text{d}y}{\text{d}x}$ of the given function
y = $x^{3} -3x^{2} + 8x $
$\frac{\text{d}y}{\text{d}x} = 3x^{2} -6x + 8 $
Now we will find the 2nd derivative
$\frac{\text{d}^{2}y}{\text{d}x^{2}}$ = 6x - 6
Example 2 : If y = sin(x) + cos(x) then find $\frac{\text{d}^{2}y}{\text{d}x^{2}}$
Solution : y = sin(x) + cos(x)
First we will find $\frac{\text{d}y}{\text{d}x}$ of the given function
y = sin(x) + cos(x)
$\frac{\text{d}y}{\text{d}x} = cos(x) -sin(x) $
Now we will find the 2nd derivative
$\frac{\text{d}^{2}y}{\text{d}x^{2}}$ = -sin(x) - cos(x)
Example 3 : If y = $x^{3}.sin(x) $ then find $\frac{\text{d}^{2}y}{\text{d}x^{2}}$
Solution : y = $x^{3}.sin(x) $
First we will find $\frac{\text{d}y}{\text{d}x}$ of the given function using the product rule
y = $x^{3}.sin(x) $
$\frac{\text{d}y}{\text{d}x} = \frac{\text{d}}{\text{d}x}(x^{3}).sin(x) + x^{3}.\frac{\text{d}}{\text{d}x}(sin(x))$
= $3x^{2}.sin(x) + x^{3} cos(x)$
Now we will find the 2nd derivative
$\frac{\text{d}^{2}y}{\text{d}x^{2}} = sin(x).\frac{\text{d}}{\text{d}x}(3x^{2}) + 3x^{2}.\frac{\text{d}}{\text{d}x}(sin(x))+ cos(x).\frac{\text{d}}{\text{d}x}(x^{3} ) + x^{3}\frac{\text{d}}{\text{d}x}(cos(x))$
= 6x. sin(x) + $3x^{2}.cos(x) + 3x^{2}.cos(x) + x^{3}.(-sin(x))$
$\frac{\text{d}^{2}y}{\text{d}x^{2}} = - x^{3}.sin(x) + 6x^{2}.cos(x) + 6x.sin(x)$