Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of x^-3
Derivative of x^2*sqrt(x)
Derivative of 1/2*x^2
Derivative of e^(5*x)
Identical expressions
cbrt(x*(six -x)^ two)
cubic root of (x multiply by (6 minus x) squared )
cubic root of (x multiply by (six minus x) to the power of two)
cbrt(x*(6-x)2)
cbrtx*6-x2
cbrt(x*(6-x)²)
cbrt(x*(6-x) to the power of 2)
cbrt(x(6-x)^2)
cbrt(x(6-x)2)
cbrtx6-x2
cbrtx6-x^2
Similar expressions
cbrt(x*(6+x)^2)

The derivative of the function/ cbrt(x*(6-x)^2)

Derivative of cbrt(x*(6-x)^2)

The solution

You have entered [src]

   ____________
3 /          2 
\/  x*(6 - x)

$$\sqrt[3]{x \left(6 - x\right)^{2}}$$

(x*(6 - x)^2)^(1/3)

Detail solution

Let $u = x \left(6 - x\right)^{2}$ .
Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \left(6 - x\right)^{2}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \left(6 - x\right)^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 6 - x$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(6 - x\right)$ :
    1. Differentiate $6 - x$ term by term:
      1. The derivative of the constant $6$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      The result is: $-1$
    The result of the chain rule is:
    
    $2 x - 12$
  The result is: $x \left(2 x - 12\right) + \left(6 - x\right)^{2}$
The result of the chain rule is:

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

$\frac{x^{2} - 8 x + 12}{x^{\frac{2}{3}} \left|{\left(x - 6\right)^{\frac{4}{3}}}\right|}$

The answer is:

$\frac{x^{2} - 8 x + 12}{x^{\frac{2}{3}} \left|{\left(x - 6\right)^{\frac{4}{3}}}\right|}$

The first derivative [src]

                  /       2                \
3 ___         2/3 |(6 - x)    x*(-12 + 2*x)|
\/ x *|-6 + x|   *|-------- + -------------|
                  \   3             3      /
--------------------------------------------
                          2                 
                 x*(6 - x)

$$\frac{\sqrt[3]{x} \left|{x - 6}\right|^{\frac{2}{3}} \left(\frac{x \left(2 x - 12\right)}{3} + \frac{\left(6 - x\right)^{2}}{3}\right)}{x \left(6 - x\right)^{2}}$$

Simplify

The second derivative [src]

                                  /        2/3     3 ___             \                                                  
                                  ||-6 + x|      2*\/ x *sign(-6 + x)|                                                  
                         (-2 + x)*|----------- + --------------------|                                                  
          2/3                     |     2/3          3 __________    |             2/3                      2/3         
  |-6 + x|   *(-2 + x)            \    x             \/ |-6 + x|     /   2*|-6 + x|   *(-2 + x)   2*|-6 + x|   *(-4 + x)
- -------------------- + --------------------------------------------- - ---------------------- + ----------------------
           5/3                                3*x                             2/3                      2/3              
          x                                                                  x   *(-6 + x)            x   *(-6 + x)     
------------------------------------------------------------------------------------------------------------------------
                                                         -6 + x

$$\frac{\frac{\left(x - 2\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x - 6 \right)}}{\sqrt[3]{\left|{x - 6}\right|}} + \frac{\left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{3 x} + \frac{2 \left(x - 4\right) \left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 6\right)} - \frac{2 \left(x - 2\right) \left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 6\right)} - \frac{\left(x - 2\right) \left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{5}{3}}}}{x - 6}$$

Simplify

The third derivative [src]

                         /        2/3     3 ___             \              /          2/3   3 ___     2                                 3 ___                   \                                                                                                                   /        2/3     3 ___             \                                        /        2/3     3 ___             \                                                                                        
                         ||-6 + x|      2*\/ x *sign(-6 + x)|              |  |-6 + x|      \/ x *sign (-6 + x)     2*sign(-6 + x)    6*\/ x *DiracDelta(-6 + x)|                                                                                                                   ||-6 + x|      2*\/ x *sign(-6 + x)|                                        ||-6 + x|      2*\/ x *sign(-6 + x)|                                                                                        
                (-2 + x)*|----------- + --------------------|   2*(-2 + x)*|- ----------- - ------------------- + ----------------- + --------------------------|                                                                                                        2*(-2 + x)*|----------- + --------------------|                             2*(-4 + x)*|----------- + --------------------|                                                                                        
          2/3            |     2/3          3 __________    |              |       5/3                  4/3        2/3 3 __________          3 __________       |             2/3                      2/3                      2/3                       2/3                       |     2/3          3 __________    |                                        |     2/3          3 __________    |              2/3                                                                       
2*|-6 + x|               \    x             \/ |-6 + x|     /              \      x             |-6 + x|          x   *\/ |-6 + x|           \/ |-6 + x|        /   5*|-6 + x|   *(-2 + x)   8*|-6 + x|   *(-4 + x)   6*|-6 + x|   *(-2 + x)   10*|-6 + x|   *(-4 + x)              \    x             \/ |-6 + x|     /   2*(-2 + x)*sign(-6 + x)              \    x             \/ |-6 + x|     /   10*|-6 + x|   *(-2 + x)     4*(-2 + x)*sign(-6 + x)        4*(-4 + x)*sign(-6 + x)   
------------- - --------------------------------------------- + ------------------------------------------------------------------------------------------------- + ---------------------- - ---------------------- + ---------------------- - ----------------------- - ----------------------------------------------- - ----------------------- + ----------------------------------------------- + ----------------------- - ---------------------------- + ----------------------------
 2/3                                    2                                                                      9*x                                                             8/3                2/3         2            2/3         2              5/3                                  3*x*(-6 + x)                         5/3 3 __________                       3*x*(-6 + x)                           5/3                   2/3          3 __________      2/3          3 __________
x   *(-6 + x)                        3*x                                                                                                                                    3*x                  x   *(-6 + x)            x   *(-6 + x)            3*x   *(-6 + x)                                                           3*x   *\/ |-6 + x|                                                            3*x   *(-6 + x)       3*x   *(-6 + x)*\/ |-6 + x|    3*x   *(-6 + x)*\/ |-6 + x| 
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                           -6 + x

$$\frac{\frac{2 \left(x - 2\right) \left(\frac{6 \sqrt[3]{x} \delta\left(x - 6\right)}{\sqrt[3]{\left|{x - 6}\right|}} - \frac{\sqrt[3]{x} \operatorname{sign}^{2}{\left(x - 6 \right)}}{\left|{x - 6}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x - 6 \right)}}{x^{\frac{2}{3}} \sqrt[3]{\left|{x - 6}\right|}} - \frac{\left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{5}{3}}}\right)}{9 x} + \frac{2 \left(x - 4\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x - 6 \right)}}{\sqrt[3]{\left|{x - 6}\right|}} + \frac{\left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{3 x \left(x - 6\right)} - \frac{2 \left(x - 2\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x - 6 \right)}}{\sqrt[3]{\left|{x - 6}\right|}} + \frac{\left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{3 x \left(x - 6\right)} - \frac{\left(x - 2\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x - 6 \right)}}{\sqrt[3]{\left|{x - 6}\right|}} + \frac{\left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{3 x^{2}} + \frac{4 \left(x - 4\right) \operatorname{sign}{\left(x - 6 \right)}}{3 x^{\frac{2}{3}} \left(x - 6\right) \sqrt[3]{\left|{x - 6}\right|}} - \frac{4 \left(x - 2\right) \operatorname{sign}{\left(x - 6 \right)}}{3 x^{\frac{2}{3}} \left(x - 6\right) \sqrt[3]{\left|{x - 6}\right|}} + \frac{2 \left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 6\right)} - \frac{8 \left(x - 4\right) \left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 6\right)^{2}} + \frac{6 \left(x - 2\right) \left|{x - 6}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 6\right)^{2}} - \frac{2 \left(x - 2\right) \operatorname{sign}{\left(x - 6 \right)}}{3 x^{\frac{5}{3}} \sqrt[3]{\left|{x - 6}\right|}} - \frac{10 \left(x - 4\right) \left|{x - 6}\right|^{\frac{2}{3}}}{3 x^{\frac{5}{3}} \left(x - 6\right)} + \frac{10 \left(x - 2\right) \left|{x - 6}\right|^{\frac{2}{3}}}{3 x^{\frac{5}{3}} \left(x - 6\right)} + \frac{5 \left(x - 2\right) \left|{x - 6}\right|^{\frac{2}{3}}}{3 x^{\frac{8}{3}}}}{x - 6}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of cbrt(x*(6-x)^2)

The graph:

Piecewise:

The solution