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(two *x^ three + one)/(x- four)
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(2*x3+1)/(x-4)
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(2*x^3+1) divide by (x-4)
Similar expressions
(2*x^3-1)/(x-4)
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The derivative of the function/ (2*x^3+1)/(x-4)

Derivative of (2*x^3+1)/(x-4)

The solution

You have entered [src]

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

$$\frac{2 x^{3} + 1}{x - 4}$$

(2*x^3 + 1)/(x - 4)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x^{3} + 1$ and $g{\left(x \right)} = x - 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x^{3} + 1$ 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: $6 x^{2}$
  The result is: $6 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 4$ term by term:
  1. The derivative of the constant $-4$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     3           2
  2*x  + 1    6*x 
- -------- + -----
         2   x - 4
  (x - 4)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (2*x^3+1)/(x-4)

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The solution