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y=ln(sqrt((x^3)+4))

Derivative of y=ln(sqrt((x^3)+4))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   ________\
   |  /  3     |
log\\/  x  + 4 /
$$\log{\left(\sqrt{x^{3} + 4} \right)}$$
  /   /   ________\\
d |   |  /  3     ||
--\log\\/  x  + 4 //
dx                  
$$\frac{d}{d x} \log{\left(\sqrt{x^{3} + 4} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
      2   
   3*x    
----------
  / 3    \
2*\x  + 4/
$$\frac{3 x^{2}}{2 \left(x^{3} + 4\right)}$$
The second derivative [src]
    /          3   \
    |       3*x    |
3*x*|1 - ----------|
    |      /     3\|
    \    2*\4 + x //
--------------------
            3       
       4 + x        
$$\frac{3 x \left(- \frac{3 x^{3}}{2 \left(x^{3} + 4\right)} + 1\right)}{x^{3} + 4}$$
The third derivative [src]
  /        3          6  \
  |     9*x        9*x   |
3*|1 - ------ + ---------|
  |         3           2|
  |    4 + x    /     3\ |
  \             \4 + x / /
--------------------------
               3          
          4 + x           
$$\frac{3 \cdot \left(\frac{9 x^{6}}{\left(x^{3} + 4\right)^{2}} - \frac{9 x^{3}}{x^{3} + 4} + 1\right)}{x^{3} + 4}$$
The graph
Derivative of y=ln(sqrt((x^3)+4))