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The derivative of the function/ (3-4*x^2)/(x+2)

Derivative of (3-4*x^2)/(x+2)

The solution

You have entered [src]

       2
3 - 4*x 
--------
 x + 2

$$\frac{3 - 4 x^{2}}{x + 2}$$

  /       2\
d |3 - 4*x |
--|--------|
dx\ x + 2  /

$$\frac{d}{d x} \frac{3 - 4 x^{2}}{x + 2}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = 3 - 4 x^{2}$ and $g{\left(x \right)} = x + 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $3 - 4 x^{2}$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 8 x$
  The result is: $- 8 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{4 x^{2} - 8 x \left(x + 2\right) - 3}{\left(x + 2\right)^{2}}$

The answer is:

$\frac{4 x^{2} - 8 x \left(x + 2\right) - 3}{\left(x + 2\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2        
  3 - 4*x     8*x 
- -------- - -----
         2   x + 2
  (x + 2)

$$- \frac{8 x}{x + 2} - \frac{3 - 4 x^{2}}{\left(x + 2\right)^{2}}$$

Simplify

The second derivative [src]

  /             2        \
  |     -3 + 4*x     8*x |
2*|-4 - --------- + -----|
  |             2   2 + x|
  \      (2 + x)         /
--------------------------
          2 + x

$$\frac{2 \cdot \left(\frac{8 x}{x + 2} - 4 - \frac{4 x^{2} - 3}{\left(x + 2\right)^{2}}\right)}{x + 2}$$

Simplify

3-я производная [src]

  /            2        \
  |    -3 + 4*x     8*x |
6*|4 + --------- - -----|
  |            2   2 + x|
  \     (2 + x)         /
-------------------------
                2        
         (2 + x)

$$\frac{6 \left(- \frac{8 x}{x + 2} + 4 + \frac{4 x^{2} - 3}{\left(x + 2\right)^{2}}\right)}{\left(x + 2\right)^{2}}$$

Simplify

The third derivative [src]

  /            2        \
  |    -3 + 4*x     8*x |
6*|4 + --------- - -----|
  |            2   2 + x|
  \     (2 + x)         /
-------------------------
                2        
         (2 + x)

$$\frac{6 \left(- \frac{8 x}{x + 2} + 4 + \frac{4 x^{2} - 3}{\left(x + 2\right)^{2}}\right)}{\left(x + 2\right)^{2}}$$

Simplify

The graph

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Derivative of (3-4*x^2)/(x+2)

The graph:

Piecewise:

The solution