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

Derivative of (2x)/(3x+1)

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. 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: $2$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   /       3*x  \
12*|-1 + -------|
   \     1 + 3*x/
-----------------
             2   
    (1 + 3*x)

$$\frac{12 \left(\frac{3 x}{3 x + 1} - 1\right)}{\left(3 x + 1\right)^{2}}$$

Simplify

The third derivative [src]

    /      3*x  \
108*|1 - -------|
    \    1 + 3*x/
-----------------
             3   
    (1 + 3*x)

$$\frac{108 \left(- \frac{3 x}{3 x + 1} + 1\right)}{\left(3 x + 1\right)^{3}}$$

Simplify