# Second Order Derivatives

**Covid-19 has led the world to go through a phenomenal transition .**

**E-learning is the future today.**

**Stay Home , Stay Safe and keep learning!!!**

To find the 2nd derivative of the given function first find the 1st derivative of the given function with respect to x and then again find the derivative of it with respect to x.

## Examples on Second Order Derivatives

**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) + $3x^{2}.cos(x) + 3x^{2}.cos(x) + x^{3}.(-sin(x))$

$\frac{\text{d}^{2}y}{\text{d}x^{2}} = - x^{3}.sin(x) + 6x^{2}.cos(x) + 6x.sin(x)$

**Note :**f "(x) < 0 $\Rightarrow $ x = c is a point of local maximum.

f "(x) > 0 $\Rightarrow $ x = c is a point of local minimum.

**12th grade math**

Home

Home

**Covid-19 has affected physical interactions between people.**

**Don't let it affect your learning.**