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Derivative of -x^4
Derivative of x^2*cos(2*x)
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Derivative of x^2/(x^2+3)
Identical expressions
(two *x^ seven + fourteen *x^ three - five *x+ one)/sqrt(three *x+ one)
(2 multiply by x to the power of 7 plus 14 multiply by x cubed minus 5 multiply by x plus 1) divide by square root of (3 multiply by x plus 1)
(two multiply by x to the power of seven plus fourteen multiply by x to the power of three minus five multiply by x plus one) divide by square root of (three multiply by x plus one)
(2*x^7+14*x^3-5*x+1)/√(3*x+1)
(2*x7+14*x3-5*x+1)/sqrt(3*x+1)
2*x7+14*x3-5*x+1/sqrt3*x+1
(2*x⁷+14*x³-5*x+1)/sqrt(3*x+1)
(2*x to the power of 7+14*x to the power of 3-5*x+1)/sqrt(3*x+1)
(2x^7+14x^3-5x+1)/sqrt(3x+1)
(2x7+14x3-5x+1)/sqrt(3x+1)
2x7+14x3-5x+1/sqrt3x+1
2x^7+14x^3-5x+1/sqrt3x+1
(2*x^7+14*x^3-5*x+1) divide by sqrt(3*x+1)
Similar expressions
(2*x^7+14*x^3-5*x-1)/sqrt(3*x+1)
(2*x^7+14*x^3-5*x+1)/sqrt(3*x-1)
(2*x^7+14*x^3+5*x+1)/sqrt(3*x+1)
(2*x^7-14*x^3-5*x+1)/sqrt(3*x+1)

The derivative of the function/ (2*x^7+14*x^3-5*x+1)/sqrt(3*x+1)

Derivative of (2x^7+14x^3-5x+1)/sqrt(3x+1)

The solution

You have entered [src]

   7       3          
2*x  + 14*x  - 5*x + 1
----------------------
       _________      
     \/ 3*x + 1

$$\frac{\left(- 5 x + \left(2 x^{7} + 14 x^{3}\right)\right) + 1}{\sqrt{3 x + 1}}$$

(2*x^7 + 14*x^3 - 5*x + 1)/sqrt(3*x + 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x^{7} + 14 x^{3} - 5 x + 1$ and $g{\left(x \right)} = \sqrt{3 x + 1}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x^{7} + 14 x^{3} - 5 x + 1$ 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: $x$ goes to $1$
    So, the result is: $-5$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$
    So, the result is: $14 x^{6}$
  4. 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: $42 x^{2}$
  The result is: $14 x^{6} + 42 x^{2} - 5$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 1\right)$ :
  1. Differentiate $3 x + 1$ 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: $x$ goes to $1$
      So, the result is: $3$
    The result is: $3$
  The result of the chain rule is:
  
  $\frac{3}{2 \sqrt{3 x + 1}}$
Now plug in to the quotient rule:

$\frac{\sqrt{3 x + 1} \left(14 x^{6} + 42 x^{2} - 5\right) - \frac{3 \left(2 x^{7} + 14 x^{3} - 5 x + 1\right)}{2 \sqrt{3 x + 1}}}{3 x + 1}$
Now simplify:

$\frac{78 x^{7} + 28 x^{6} + 210 x^{3} + 84 x^{2} - 15 x - 13}{2 \left(3 x + 1\right)^{\frac{3}{2}}}$

The answer is:

$\frac{78 x^{7} + 28 x^{6} + 210 x^{3} + 84 x^{2} - 15 x - 13}{2 \left(3 x + 1\right)^{\frac{3}{2}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         6       2     /   7       3          \
-5 + 14*x  + 42*x    3*\2*x  + 14*x  - 5*x + 1/
------------------ - --------------------------
     _________                        3/2      
   \/ 3*x + 1              2*(3*x + 1)

$$\frac{14 x^{6} + 42 x^{2} - 5}{\sqrt{3 x + 1}} - \frac{3 \left(\left(- 5 x + \left(2 x^{7} + 14 x^{3}\right)\right) + 1\right)}{2 \left(3 x + 1\right)^{\frac{3}{2}}}$$

Simplify

The second derivative [src]

  /           6       2                     /             7       3\\
  |  -5 + 14*x  + 42*x         /     4\   9*\1 - 5*x + 2*x  + 14*x /|
3*|- ------------------ + 28*x*\1 + x / + --------------------------|
  |       1 + 3*x                                           2       |
  \                                              4*(1 + 3*x)        /
---------------------------------------------------------------------
                               _________                             
                             \/ 1 + 3*x

$$\frac{3 \left(28 x \left(x^{4} + 1\right) - \frac{14 x^{6} + 42 x^{2} - 5}{3 x + 1} + \frac{9 \left(2 x^{7} + 14 x^{3} - 5 x + 1\right)}{4 \left(3 x + 1\right)^{2}}\right)}{\sqrt{3 x + 1}}$$

Simplify

The third derivative [src]

  /                  /             7       3\      /         6       2\         /     4\\
  |          4   135*\1 - 5*x + 2*x  + 14*x /   27*\-5 + 14*x  + 42*x /   126*x*\1 + x /|
3*|28 + 140*x  - ---------------------------- + ----------------------- - --------------|
  |                                 3                            2           1 + 3*x    |
  \                      8*(1 + 3*x)                  4*(1 + 3*x)                       /
-----------------------------------------------------------------------------------------
                                         _________                                       
                                       \/ 1 + 3*x

$$\frac{3 \left(140 x^{4} - \frac{126 x \left(x^{4} + 1\right)}{3 x + 1} + 28 + \frac{27 \left(14 x^{6} + 42 x^{2} - 5\right)}{4 \left(3 x + 1\right)^{2}} - \frac{135 \left(2 x^{7} + 14 x^{3} - 5 x + 1\right)}{8 \left(3 x + 1\right)^{3}}\right)}{\sqrt{3 x + 1}}$$

Simplify

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Derivative of (2x^7+14x^3-5x+1)/sqrt(3x+1)

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

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Derivative of (2*x^7+14*x^3-5*x+1)/sqrt(3*x+1)

The graph:

Piecewise:

The solution

Derivative of (2x^7+14x^3-5x+1)/sqrt(3x+1)