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Identical expressions
(ln(three x^ four))/(7x^3- nine)
(ln(3x to the power of 4)) divide by (7x cubed minus 9)
(ln(three x to the power of four)) divide by (7x cubed minus nine)
(ln(3x4))/(7x3-9)
ln3x4/7x3-9
(ln(3x⁴))/(7x³-9)
(ln(3x to the power of 4))/(7x to the power of 3-9)
ln3x^4/7x^3-9
(ln(3x^4)) divide by (7x^3-9)
Similar expressions
(ln(3x^4))/(7x^3+9)

The derivative of the function/ (ln(3x^4))/(7x^3-9)

Derivative of (ln(3x^4))/(7x^3-9)

The solution

You have entered [src]

   /   4\
log\3*x /
---------
    3    
 7*x  - 9

$$\frac{\log{\left(3 x^{4} \right)}}{7 x^{3} - 9}$$

  /   /   4\\
d |log\3*x /|
--|---------|
dx|    3    |
  \ 7*x  - 9/

$$\frac{d}{d x} \frac{\log{\left(3 x^{4} \right)}}{7 x^{3} - 9}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(3 x^{4} \right)}$ and $g{\left(x \right)} = 7 x^{3} - 9$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 3 x^{4}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x^{4}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
    So, the result is: $12 x^{3}$
  The result of the chain rule is:
  
  $\frac{4}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $7 x^{3} - 9$ term by term:
  1. The derivative of the constant $-9$ 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: $21 x^{2}$
  The result is: $21 x^{2}$
Now plug in to the quotient rule:

$\frac{- 21 x^{2} \log{\left(3 x^{4} \right)} + \frac{4 \cdot \left(7 x^{3} - 9\right)}{x}}{\left(7 x^{3} - 9\right)^{2}}$
Now simplify:

$\frac{- 21 x^{3} \log{\left(3 x^{4} \right)} + 28 x^{3} - 36}{x \left(7 x^{3} - 9\right)^{2}}$

The answer is:

$\frac{- 21 x^{3} \log{\left(3 x^{4} \right)} + 28 x^{3} - 36}{x \left(7 x^{3} - 9\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                   2    /   4\
     4         21*x *log\3*x /
------------ - ---------------
  /   3    \               2  
x*\7*x  - 9/     /   3    \   
                 \7*x  - 9/

$$- \frac{21 x^{2} \log{\left(3 x^{4} \right)}}{\left(7 x^{3} - 9\right)^{2}} + \frac{4}{x \left(7 x^{3} - 9\right)}$$

Simplify

The second derivative [src]

  /                        /           3  \          \
  |                        |       21*x   |    /   4\|
  |                   21*x*|-1 + ---------|*log\3*x /|
  |                        |             3|          |
  |  2       84*x          \     -9 + 7*x /          |
2*|- -- - --------- + -------------------------------|
  |   2           3                      3           |
  \  x    -9 + 7*x               -9 + 7*x            /
------------------------------------------------------
                              3                       
                      -9 + 7*x

$$\frac{2 \cdot \left(\frac{21 x \left(\frac{21 x^{3}}{7 x^{3} - 9} - 1\right) \log{\left(3 x^{4} \right)}}{7 x^{3} - 9} - \frac{84 x}{7 x^{3} - 9} - \frac{2}{x^{2}}\right)}{7 x^{3} - 9}$$

Simplify

The third derivative [src]

  /                                           /           3            6   \          \
  |                     /           3  \      |      126*x       1323*x    |    /   4\|
  |                     |       21*x   |   21*|1 - --------- + ------------|*log\3*x /|
  |                 252*|-1 + ---------|      |            3              2|          |
  |                     |             3|      |    -9 + 7*x    /        3\ |          |
  |4       126          \     -9 + 7*x /      \                \-9 + 7*x / /          |
2*|-- + --------- + -------------------- - -------------------------------------------|
  | 3           3                3                                  3                 |
  \x    -9 + 7*x         -9 + 7*x                           -9 + 7*x                  /
---------------------------------------------------------------------------------------
                                               3                                       
                                       -9 + 7*x

$$\frac{2 \cdot \left(\frac{252 \cdot \left(\frac{21 x^{3}}{7 x^{3} - 9} - 1\right)}{7 x^{3} - 9} - \frac{21 \cdot \left(\frac{1323 x^{6}}{\left(7 x^{3} - 9\right)^{2}} - \frac{126 x^{3}}{7 x^{3} - 9} + 1\right) \log{\left(3 x^{4} \right)}}{7 x^{3} - 9} + \frac{126}{7 x^{3} - 9} + \frac{4}{x^{3}}\right)}{7 x^{3} - 9}$$

Simplify

The graph

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Derivative of (ln(3x^4))/(7x^3-9)

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