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Derivative of:
Derivative of y=1/cos²x
Derivative of y=1/2
Derivative of x/(x^4-3*x)
Derivative of cos3^x
Identical expressions
x/(x^ four - three *x)
x divide by (x to the power of 4 minus 3 multiply by x)
x divide by (x to the power of four minus three multiply by x)
x/(x4-3*x)
x/x4-3*x
x/(x⁴-3*x)
x/(x^4-3x)
x/(x4-3x)
x/x4-3x
x/x^4-3x
x divide by (x^4-3*x)
Similar expressions
x/(x^4+3*x)

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

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

The solution

You have entered [src]

   x    
--------
 4      
x  - 3*x

$$\frac{x}{x^{4} - 3 x}$$

x/(x^4 - 3*x)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{4} - 3 x$ term by term:
  1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
  2. 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: $-3$
  The result is: $4 x^{3} - 3$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /                                  3              2\
   |        /        3\    /        3\    /        3\ |
   |     12*\-3 + 4*x /    \-3 + 4*x /    \-3 + 4*x / |
-6*|10 - -------------- + ------------- - ------------|
   |              3                   2    3 /      3\|
   |        -3 + x         3 /      3\    x *\-3 + x /|
   \                      x *\-3 + x /                /
-------------------------------------------------------
                                2                      
                       /      3\                       
                       \-3 + x /

$$- \frac{6 \left(10 - \frac{12 \left(4 x^{3} - 3\right)}{x^{3} - 3} - \frac{\left(4 x^{3} - 3\right)^{2}}{x^{3} \left(x^{3} - 3\right)} + \frac{\left(4 x^{3} - 3\right)^{3}}{x^{3} \left(x^{3} - 3\right)^{2}}\right)}{\left(x^{3} - 3\right)^{2}}$$

Simplify

The graph

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

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Piecewise:

The solution