Mister Exam

Other calculators

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

Derivative of x^2/(1+x^3)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   2  
  x   
------
     3
1 + x
x2x3+1\frac{x^{2}}{x^{3} + 1} 
x^2/(1 + x^3)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x2f{\left(x \right)} = x^{2} and g(x)=x3+1g{\left(x \right)} = x^{3} + 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x2x^{2} goes to 2x2 x

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x3+1x^{3} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      The result is: 3x23 x^{2}

    Now plug in to the quotient rule:

    −3x4+2x(x3+1)(x3+1)2\frac{- 3 x^{4} + 2 x \left(x^{3} + 1\right)}{\left(x^{3} + 1\right)^{2}}

  2. Now simplify:

    x(2−x3)(x3+1)2\frac{x \left(2 - x^{3}\right)}{\left(x^{3} + 1\right)^{2}}

The answer is:

x(2−x3)(x3+1)2\frac{x \left(2 - x^{3}\right)}{\left(x^{3} + 1\right)^{2}}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        4           
     3*x       2*x  
- --------- + ------
          2        3
  /     3\    1 + x 
  \1 + x /
−3x4(x3+1)2+2xx3+1- \frac{3 x^{4}}{\left(x^{3} + 1\right)^{2}} + \frac{2 x}{x^{3} + 1}
Simplify
The second derivative [src] 
  /                  /         3 \\
  |                3 |      3*x  ||
  |             3*x *|-1 + ------||
  |        3         |          3||
  |     6*x          \     1 + x /|
2*|1 - ------ + ------------------|
  |         3              3      |
  \    1 + x          1 + x       /
-----------------------------------
                    3              
               1 + x
2(3x3(3x3x3+1−1)x3+1−6x3x3+1+1)x3+1\frac{2 \left(\frac{3 x^{3} \left(\frac{3 x^{3}}{x^{3} + 1} - 1\right)}{x^{3} + 1} - \frac{6 x^{3}}{x^{3} + 1} + 1\right)}{x^{3} + 1}
Simplify
3-я производная [src] 
     /            6         3 \
   2 |        27*x      36*x  |
6*x *|-10 - --------- + ------|
     |              2        3|
     |      /     3\    1 + x |
     \      \1 + x /          /
-------------------------------
                   2           
           /     3\            
           \1 + x /
6x2(−27x6(x3+1)2+36x3x3+1−10)(x3+1)2\frac{6 x^{2} \left(- \frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} + \frac{36 x^{3}}{x^{3} + 1} - 10\right)}{\left(x^{3} + 1\right)^{2}}
Simplify
The third derivative [src] 
     /            6         3 \
   2 |        27*x      36*x  |
6*x *|-10 - --------- + ------|
     |              2        3|
     |      /     3\    1 + x |
     \      \1 + x /          /
-------------------------------
                   2           
           /     3\            
           \1 + x /
6x2(−27x6(x3+1)2+36x3x3+1−10)(x3+1)2\frac{6 x^{2} \left(- \frac{27 x^{6}}{\left(x^{3} + 1\right)^{2}} + \frac{36 x^{3}}{x^{3} + 1} - 10\right)}{\left(x^{3} + 1\right)^{2}}
Simplify
    © Mister Exam – Calculator