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Identical expressions
y=log4(x+x^ three)
y equally logarithm of 4(x plus x cubed )
y equally logarithm of 4(x plus x to the power of three)
y=log4(x+x3)
y=log4x+x3
y=log4(x+x³)
y=log4(x+x to the power of 3)
y=log4x+x^3
Similar expressions
y=log4(x-x^3)

The derivative of the function/ y=log4(x+x^3)

Derivative of y=log4(x+x^3)

The solution

You have entered [src]

   /     3\
log\x + x /
-----------
   log(4)

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

  /   /     3\\
d |log\x + x /|
--|-----------|
dx\   log(4)  /

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           2   
    1 + 3*x    
---------------
/     3\       
\x + x /*log(4)

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

Simplify

The second derivative [src]

              2
    /       2\ 
    \1 + 3*x / 
6 - -----------
     2 /     2\
    x *\1 + x /
---------------
/     2\       
\1 + x /*log(4)

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

Simplify

The third derivative [src]

  /                             3 \
  |      /       2\   /       2\  |
  |    9*\1 + 3*x /   \1 + 3*x /  |
2*|3 - ------------ + ------------|
  |            2                 2|
  |       1 + x        2 /     2\ |
  \                   x *\1 + x / /
-----------------------------------
           /     2\                
         x*\1 + x /*log(4)

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

Simplify

The graph

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Derivative of y=log4(x+x^3)

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The solution