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Derivative of:
Derivative of e^(3*x)
Derivative of e^x+1
Derivative of (x^2-1)/(x*3+1)
Derivative of 2/x^3
Graphing y =:
(x^2-1)/(x*3+1)
Identical expressions
(x^ two - one)/(x* three + one)
(x squared minus 1) divide by (x multiply by 3 plus 1)
(x to the power of two minus one) divide by (x multiply by three plus one)
(x2-1)/(x*3+1)
x2-1/x*3+1
(x²-1)/(x*3+1)
(x to the power of 2-1)/(x*3+1)
(x^2-1)/(x3+1)
(x2-1)/(x3+1)
x2-1/x3+1
x^2-1/x3+1
(x^2-1) divide by (x*3+1)
Similar expressions
(x^2-1)/(x*3-1)
(x^2+1)/(x*3+1)

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

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

The solution

You have entered [src]

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

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

  /  2    \
d | x  - 1|
--|-------|
dx\x*3 + 1/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

$\frac{3 x^{2} + 2 x + 3}{9 x^{2} + 6 x + 1}$

The answer is:

$\frac{3 x^{2} + 2 x + 3}{9 x^{2} + 6 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    / 2    \          
  3*\x  - 1/     2*x  
- ---------- + -------
           2   x*3 + 1
  (x*3 + 1)

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

Simplify

The second derivative [src]

  /                /      2\\
  |      6*x     9*\-1 + x /|
2*|1 - ------- + -----------|
  |    1 + 3*x             2|
  \               (1 + 3*x) /
-----------------------------
           1 + 3*x

$$\frac{2 \left(- \frac{6 x}{3 x + 1} + 1 + \frac{9 \left(x^{2} - 1\right)}{\left(3 x + 1\right)^{2}}\right)}{3 x + 1}$$

Simplify

The third derivative [src]

   /       /      2\          \
   |     9*\-1 + x /     6*x  |
18*|-1 - ----------- + -------|
   |               2   1 + 3*x|
   \      (1 + 3*x)           /
-------------------------------
                    2          
           (1 + 3*x)

$$\frac{18 \cdot \left(\frac{6 x}{3 x + 1} - 1 - \frac{9 \left(x^{2} - 1\right)}{\left(3 x + 1\right)^{2}}\right)}{\left(3 x + 1\right)^{2}}$$

Simplify

The graph

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

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The solution