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The derivative of the function/ (4x+1)/(3x-1)

Derivative of (4x+1)/(3x-1)

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $4 x + 1$ 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: $4$
  The result is: $4$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /    3*(1 + 4*x)\
54*|4 - -----------|
   \      -1 + 3*x /
--------------------
              3     
    (-1 + 3*x)

$$\frac{54 \left(4 - \frac{3 \left(4 x + 1\right)}{3 x - 1}\right)}{\left(3 x - 1\right)^{3}}$$

Simplify

The graph

Derivative of (4x+1)/(3x-1)