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Identical expressions
cbrt(x^ three + three *x^ two - one)
cubic root of (x cubed plus 3 multiply by x squared minus 1)
cubic root of (x to the power of three plus three multiply by x to the power of two minus one)
cbrt(x3+3*x2-1)
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cbrt(x^3+3x^2-1)
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cbrtx3+3x2-1
cbrtx^3+3x^2-1
Similar expressions
cbrt(x^3+3*x^2+1)
cbrt(x^3-3*x^2-1)

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

Derivative of cbrt(x^3+3*x^2-1)

The solution

You have entered [src]

   _______________
3 /  3      2     
\/  x  + 3*x  - 1

$$\sqrt[3]{\left(x^{3} + 3 x^{2}\right) - 1}$$

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

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /          2        2  \
  |         x *(2 + x)   |
2*|1 + x - --------------|
  |              3      2|
  \        -1 + x  + 3*x /
--------------------------
                   2/3    
   /      3      2\       
   \-1 + x  + 3*x /

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

Simplify

The third derivative [src]

  /         3        3                        \
  |      5*x *(2 + x)      6*x*(1 + x)*(2 + x)|
2*|1 + ----------------- - -------------------|
  |                    2            3      2  |
  |    /      3      2\       -1 + x  + 3*x   |
  \    \-1 + x  + 3*x /                       /
-----------------------------------------------
                              2/3              
              /      3      2\                 
              \-1 + x  + 3*x /

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

Simplify

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Derivative of cbrt(x^3+3*x^2-1)

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The solution