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Identical expressions
log(x)/(log(two)*x^ three)
logarithm of (x) divide by ( logarithm of (2) multiply by x cubed )
logarithm of (x) divide by ( logarithm of (two) multiply by x to the power of three)
log(x)/(log(2)*x3)
logx/log2*x3
log(x)/(log(2)*x³)
log(x)/(log(2)*x to the power of 3)
log(x)/(log(2)x^3)
log(x)/(log(2)x3)
logx/log2x3
logx/log2x^3
log(x) divide by (log(2)*x^3)

The derivative of the function/ log(x)/(log(2)*x^3)

Derivative of log(x)/(log(2)*x^3)

The solution

You have entered [src]

  log(x) 
---------
        3
log(2)*x

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

log(x)/((log(2)*x^3))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = x^{3} \log{\left(2 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. 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: $3 x^{2} \log{\left(2 \right)}$
Now plug in to the quotient rule:

$\frac{- 3 x^{2} \log{\left(2 \right)} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}}{x^{6} \log{\left(2 \right)}^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

-7 + 12*log(x)
--------------
   5          
  x *log(2)

$$\frac{12 \log{\left(x \right)} - 7}{x^{5} \log{\left(2 \right)}}$$

Simplify

The third derivative [src]

47 - 60*log(x)
--------------
   6          
  x *log(2)

$$\frac{47 - 60 \log{\left(x \right)}}{x^{6} \log{\left(2 \right)}}$$

Simplify

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Identical expressions

Derivative of log(x)/(log(2)*x^3)

The graph:

Piecewise:

The solution