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Derivative of:
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Identical expressions
(four *x+ one)/(x+ three)
(4 multiply by x plus 1) divide by (x plus 3)
(four multiply by x plus one) divide by (x plus three)
(4x+1)/(x+3)
4x+1/x+3
(4*x+1) divide by (x+3)
Similar expressions
(4*x+1)/(x-3)
(4*x-1)/(x+3)

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

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

The solution

You have entered [src]

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

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

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

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 $g{\left(x \right)} = x + 3$ .

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 $x + 3$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{11}{\left(x + 3\right)^{2}}$

The answer is:

$\frac{11}{\left(x + 3\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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