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Derivative of:
Derivative of -x^4
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Identical expressions
(one +x-4sqrtx)/x
(1 plus x minus 4 square root of x) divide by x
(one plus x minus 4 square root of x) divide by x
(1+x-4√x)/x
1+x-4sqrtx/x
(1+x-4sqrtx) divide by x
Similar expressions
(1-x-4sqrtx)/x
(1+x+4sqrtx)/x

The derivative of the function/ (1+x-4sqrtx)/x

Derivative of (1+x-4sqrtx)/x

The solution

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            ___
1 + x - 4*\/ x 
---------------
       x

$$\frac{- 4 \sqrt{x} + \left(x + 1\right)}{x}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 4 \sqrt{x} + x + 1$ and $g{\left(x \right)} = x$ .

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

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

$\frac{2 \sqrt{x} - 1}{x^{2}}$

The answer is:

$\frac{2 \sqrt{x} - 1}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

         /      2  \                      
       2*|1 - -----|                      
         |      ___|     /            ___\
 1       \    \/ x /   2*\1 + x - 4*\/ x /
---- - ------------- + -------------------
 5/2          2                  3        
x            x                  x

$$- \frac{2 \left(1 - \frac{2}{\sqrt{x}}\right)}{x^{2}} + \frac{2 \left(- 4 \sqrt{x} + x + 1\right)}{x^{3}} + \frac{1}{x^{\frac{5}{2}}}$$

Simplify

The third derivative [src]

  /                                   /      2  \\
  |                                 2*|1 - -----||
  |             /            ___\     |      ___||
  |    3      2*\1 + x - 4*\/ x /     \    \/ x /|
3*|- ------ - ------------------- + -------------|
  |     7/2             4                  3     |
  \  2*x               x                  x      /

$$3 \left(\frac{2 \left(1 - \frac{2}{\sqrt{x}}\right)}{x^{3}} - \frac{2 \left(- 4 \sqrt{x} + x + 1\right)}{x^{4}} - \frac{3}{2 x^{\frac{7}{2}}}\right)$$

Simplify

The graph

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Derivative of (1+x-4sqrtx)/x

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The solution