Mister Exam

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Derivative of sqrt(x^3)

You have entered [src]

   ____
  /  3 
\/  x

$$\sqrt{x^{3}}$$

sqrt(x^3)

Detail solution

Let $u = x^{3}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{3}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
The result of the chain rule is:

$\frac{3 x^{2}}{2 \sqrt{x^{3}}}$

The answer is:

$\frac{3 x^{2}}{2 \sqrt{x^{3}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     ____
    /  3 
3*\/  x  
---------
   2*x

$$\frac{3 \sqrt{x^{3}}}{2 x}$$

Simplify

The second derivative [src]

     ____
    /  3 
3*\/  x  
---------
      2  
   4*x

$$\frac{3 \sqrt{x^{3}}}{4 x^{2}}$$

Simplify

The third derivative [src]

      ____
     /  3 
-3*\/  x  
----------
      3   
   8*x

$$- \frac{3 \sqrt{x^{3}}}{8 x^{3}}$$

Simplify

The graph