Mister Exam

Other calculators

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

Derivative of cbrt(x^3+1)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   ________
3 /  3     
\/  x  + 1
x3+13\sqrt[3]{x^{3} + 1} 
(x^3 + 1)^(1/3)
Detail solution

  1. Let u=x3+1u = x^{3} + 1.

  2. Apply the power rule: u3\sqrt[3]{u} goes to 13u23\frac{1}{3 u^{\frac{2}{3}}}

  3. Then, apply the chain rule. Multiply by ddx(x3+1)\frac{d}{d x} \left(x^{3} + 1\right):

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

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

      2. The derivative of the constant 11 is zero.

      The result is: 3x23 x^{2}

    The result of the chain rule is:

    x2(x3+1)23\frac{x^{2}}{\left(x^{3} + 1\right)^{\frac{2}{3}}}

  4. Now simplify:

    x2(x3+1)23\frac{x^{2}}{\left(x^{3} + 1\right)^{\frac{2}{3}}}

The answer is:

x2(x3+1)23\frac{x^{2}}{\left(x^{3} + 1\right)^{\frac{2}{3}}}

The graph
02468-8-6-4-2-1010020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      2    
     x     
-----------
        2/3
/ 3    \   
\x  + 1/
x2(x3+1)23\frac{x^{2}}{\left(x^{3} + 1\right)^{\frac{2}{3}}}
Simplify
The second derivative [src] 
    /       3  \
    |      x   |
2*x*|1 - ------|
    |         3|
    \    1 + x /
----------------
          2/3   
  /     3\      
  \1 + x /
2x(−x3x3+1+1)(x3+1)23\frac{2 x \left(- \frac{x^{3}}{x^{3} + 1} + 1\right)}{\left(x^{3} + 1\right)^{\frac{2}{3}}}
Simplify
The third derivative [src] 
  /        3          6  \
  |     6*x        5*x   |
2*|1 - ------ + ---------|
  |         3           2|
  |    1 + x    /     3\ |
  \             \1 + x / /
--------------------------
               2/3        
       /     3\           
       \1 + x /
2(5x6(x3+1)2−6x3x3+1+1)(x3+1)23\frac{2 \left(\frac{5 x^{6}}{\left(x^{3} + 1\right)^{2}} - \frac{6 x^{3}}{x^{3} + 1} + 1\right)}{\left(x^{3} + 1\right)^{\frac{2}{3}}}
Simplify
The graph
Derivative of cbrt(x^3+1)
    © Mister Exam – Calculator