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

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

The solution

You have entered [src]

x + 2
-----
x + 3

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

(x + 2)/(x + 3)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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