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log(four *x)/(one -x^ three)
logarithm of (4 multiply by x) divide by (1 minus x cubed )
logarithm of (four multiply by x) divide by (one minus x to the power of three)
log(4*x)/(1-x3)
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log(4x)/(1-x^3)
log(4x)/(1-x3)
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log(4*x) divide by (1-x^3)
Similar expressions
log(4*x)/(1+x^3)

The derivative of the function/ log(4*x)/(1-x^3)

Derivative of log(4*x)/(1-x^3)

The solution

You have entered [src]

log(4*x)
--------
      3 
 1 - x

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

log(4*x)/(1 - x^3)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(4 x \right)}$ and $g{\left(x \right)} = 1 - x^{3}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                       /         3          6   \         
                    /          3 \     |     18*x       27*x    |         
                    |       3*x  |   6*|1 - ------- + ----------|*log(4*x)
                 18*|-1 + -------|     |          3            2|         
                    |           3|     |    -1 + x    /      3\ |         
     9      2       \     -1 + x /     \              \-1 + x / /         
- ------- - -- - ----------------- + -------------------------------------
        3    3              3                             3               
  -1 + x    x         -1 + x                        -1 + x                
--------------------------------------------------------------------------
                                       3                                  
                                 -1 + x

$$\frac{- \frac{18 \left(\frac{3 x^{3}}{x^{3} - 1} - 1\right)}{x^{3} - 1} + \frac{6 \left(\frac{27 x^{6}}{\left(x^{3} - 1\right)^{2}} - \frac{18 x^{3}}{x^{3} - 1} + 1\right) \log{\left(4 x \right)}}{x^{3} - 1} - \frac{9}{x^{3} - 1} - \frac{2}{x^{3}}}{x^{3} - 1}$$

Simplify

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Derivative of log(4*x)/(1-x^3)

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The solution