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Derivative of 4*x^4
Identical expressions
y=ln(sqrt((x^ three)+ four))
y equally ln( square root of ((x cubed ) plus 4))
y equally ln( square root of ((x to the power of three) plus four))
y=ln(√((x^3)+4))
y=ln(sqrt((x3)+4))
y=lnsqrtx3+4
y=ln(sqrt((x³)+4))
y=ln(sqrt((x to the power of 3)+4))
y=lnsqrtx^3+4
Similar expressions
y=ln(sqrt((x^3)-4))

The derivative of the function/ y=ln(sqrt((x^3)+4))

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

The solution

You have entered [src]

   /   ________\
   |  /  3     |
log\\/  x  + 4 /

$$\log{\left(\sqrt{x^{3} + 4} \right)}$$

  /   /   ________\\
d |   |  /  3     ||
--\log\\/  x  + 4 //
dx

$$\frac{d}{d x} \log{\left(\sqrt{x^{3} + 4} \right)}$$

Transform

Detail solution

Let $u = \sqrt{x^{3} + 4}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x^{3} + 4}$ :
1. Let $u = x^{3} + 4$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{3} + 4\right)$ :
  1. Differentiate $x^{3} + 4$ term by term:
    1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
    2. The derivative of the constant $4$ is zero.
    The result is: $3 x^{2}$
  The result of the chain rule is:
  
  $\frac{3 x^{2}}{2 \sqrt{x^{3} + 4}}$
The result of the chain rule is:

$\frac{3 x^{2}}{2 \left(x^{3} + 4\right)}$
Now simplify:

$\frac{3 x^{2}}{2 \left(x^{3} + 4\right)}$

The answer is:

$\frac{3 x^{2}}{2 \left(x^{3} + 4\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$\frac{3 x^{2}}{2 \left(x^{3} + 4\right)}$$

Simplify

The second derivative [src]

    /          3   \
    |       3*x    |
3*x*|1 - ----------|
    |      /     3\|
    \    2*\4 + x //
--------------------
            3       
       4 + x

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

Simplify

The third derivative [src]

  /        3          6  \
  |     9*x        9*x   |
3*|1 - ------ + ---------|
  |         3           2|
  |    4 + x    /     3\ |
  \             \4 + x / /
--------------------------
               3          
          4 + x

$$\frac{3 \cdot \left(\frac{9 x^{6}}{\left(x^{3} + 4\right)^{2}} - \frac{9 x^{3}}{x^{3} + 4} + 1\right)}{x^{3} + 4}$$

Simplify

The graph

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

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The solution