second order derivatives |ab calculus bc,12th grade math

# Second Order Derivatives

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Second order derivative is used in application of derivative or for graphing any polynomial , rational equations etc. If y = f(x) is a function in x then 2nd order derivative is denoted as f "(x) or $\frac{\text{d}^{2}y}{\text{d}x^{2}}$.
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.