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$((x^6)+4x^3)\(((x^3)+1)^2)$

How to use it?
Derivative of:
Derivative of 2e^x
Derivative of x/(x+1)
Derivative of x^(x^2)
Derivative of x/log(x)
Identical expressions
((x^ six)+4x^ three)\(((x^ three)+ one)^ two)
((x to the power of 6) plus 4x cubed )\(((x cubed ) plus 1) squared )
((x to the power of six) plus 4x to the power of three)\(((x to the power of three) plus one) to the power of two)
((x6)+4x3)\(((x3)+1)2)
x6+4x3\x3+12
((x⁶)+4x³)\(((x³)+1)²)
((x to the power of 6)+4x to the power of 3)\(((x to the power of 3)+1) to the power of 2)
x^6+4x^3\x^3+1^2
Similar expressions
((x^6)+4x^3)\(((x^3)-1)^2)
((x^6)-4x^3)\(((x^3)+1)^2)

The derivative of the function/ ((x^6)+4x^3)\(((x^3)+1)^2)

Derivative of ((x^6)+4x^3)\(((x^3)+1)^2)

The solution

You have entered [src]

 6      3
x  + 4*x 
---------
        2
/ 3    \ 
\x  + 1/

$$\frac{x^{6} + 4 x^{3}}{\left(x^{3} + 1\right)^{2}}$$

  / 6      3\
d |x  + 4*x |
--|---------|
dx|        2|
  |/ 3    \ |
  \\x  + 1/ /

$$\frac{d}{d x} \frac{x^{6} + 4 x^{3}}{\left(x^{3} + 1\right)^{2}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{6} + 4 x^{3}$ and $g{\left(x \right)} = \left(x^{3} + 1\right)^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{6} + 4 x^{3}$ term by term:
  1. Apply the power rule: $x^{6}$ goes to $6 x^{5}$
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    So, the result is: $12 x^{2}$
  The result is: $6 x^{5} + 12 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{3} + 1$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{3} + 1\right)$ :
  1. Differentiate $x^{3} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    The result is: $3 x^{2}$
  The result of the chain rule is:
  
  $3 x^{2} \cdot \left(2 x^{3} + 2\right)$
Now plug in to the quotient rule:

$\frac{- 3 x^{2} \cdot \left(2 x^{3} + 2\right) \left(x^{6} + 4 x^{3}\right) + \left(x^{3} + 1\right)^{2} \cdot \left(6 x^{5} + 12 x^{2}\right)}{\left(x^{3} + 1\right)^{4}}$
Now simplify:

$\frac{6 x^{2} \cdot \left(2 - x^{3}\right)}{x^{9} + 3 x^{6} + 3 x^{3} + 1}$

The answer is:

$\frac{6 x^{2} \cdot \left(2 - x^{3}\right)}{x^{9} + 3 x^{6} + 3 x^{3} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   5       2      2 / 6      3\
6*x  + 12*x    6*x *\x  + 4*x /
------------ - ----------------
         2                3    
 / 3    \         / 3    \     
 \x  + 1/         \x  + 1/

$$- \frac{6 x^{2} \left(x^{6} + 4 x^{3}\right)}{\left(x^{3} + 1\right)^{3}} + \frac{6 x^{5} + 12 x^{2}}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The second derivative [src]

    /                               /         3 \         \
    |                             3 |      9*x  | /     3\|
    |                            x *|-2 + ------|*\4 + x /|
    |               3 /     3\      |          3|         |
    |       3   12*x *\2 + x /      \     1 + x /         |
6*x*|4 + 5*x  - -------------- + -------------------------|
    |                    3                      3         |
    \               1 + x                  1 + x          /
-----------------------------------------------------------
                                 2                         
                         /     3\                          
                         \1 + x /

$$\frac{6 x \left(\frac{x^{3} \left(x^{3} + 4\right) \left(\frac{9 x^{3}}{x^{3} + 1} - 2\right)}{x^{3} + 1} + 5 x^{3} - \frac{12 x^{3} \left(x^{3} + 2\right)}{x^{3} + 1} + 4\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The third derivative [src]

   /                                          /        3          6  \                              \
   |                               3 /     3\ |    27*x       54*x   |        /         3 \         |
   |                              x *\4 + x /*|1 - ------ + ---------|      3 |      9*x  | /     3\|
   |                                          |         3           2|   9*x *|-2 + ------|*\2 + x /|
   |               3 /       3\               |    1 + x    /     3\ |        |          3|         |
   |        3   9*x *\4 + 5*x /               \             \1 + x / /        \     1 + x /         |
12*|2 + 10*x  - --------------- - ------------------------------------ + ---------------------------|
   |                      3                           3                                  3          |
   \                 1 + x                       1 + x                              1 + x           /
-----------------------------------------------------------------------------------------------------
                                                      2                                              
                                              /     3\                                               
                                              \1 + x /

$$\frac{12 \cdot \left(\frac{9 x^{3} \left(x^{3} + 2\right) \left(\frac{9 x^{3}}{x^{3} + 1} - 2\right)}{x^{3} + 1} - \frac{x^{3} \left(x^{3} + 4\right) \left(\frac{54 x^{6}}{\left(x^{3} + 1\right)^{2}} - \frac{27 x^{3}}{x^{3} + 1} + 1\right)}{x^{3} + 1} + 10 x^{3} - \frac{9 x^{3} \cdot \left(5 x^{3} + 4\right)}{x^{3} + 1} + 2\right)}{\left(x^{3} + 1\right)^{2}}$$

Simplify

The graph

$Derivative of ((x^6)+4x^3)\(((x^3)+1)^2)$

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How to use it?

Identical expressions

Similar expressions

Derivative of ((x^6)+4x^3)\(((x^3)+1)^2)

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Piecewise:

The solution