Derivative of (x/(x+5))^2

You have entered [src]

       2
/  x  \ 
|-----| 
\x + 5/

$$\left(\frac{x}{x + 5}\right)^{2}$$

(x/(x + 5))^2

Detail solution

Let $u = \frac{x}{x + 5}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{x + 5}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ and $g{\left(x \right)} = x + 5$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x + 5$ term by term:
    1. The derivative of the constant $5$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $\frac{5}{\left(x + 5\right)^{2}}$
The result of the chain rule is:

$\frac{10 x}{\left(x + 5\right) \left(x + 5\right)^{2}}$
Now simplify:

$\frac{10 x}{\left(x + 5\right)^{3}}$

The answer is:

$\frac{10 x}{\left(x + 5\right)^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2                              
   x             /  2       2*x   \
--------*(x + 5)*|----- - --------|
       2         |x + 5          2|
(x + 5)          \        (x + 5) /
-----------------------------------
                 x

$$\frac{\frac{x^{2}}{\left(x + 5\right)^{2}} \left(x + 5\right) \left(- \frac{2 x}{\left(x + 5\right)^{2}} + \frac{2}{x + 5}\right)}{x}$$

Simplify

The second derivative [src]

  /       x  \ /      3*x \
2*|-1 + -----|*|-1 + -----|
  \     5 + x/ \     5 + x/
---------------------------
                 2         
          (5 + x)

$$\frac{2 \left(\frac{x}{x + 5} - 1\right) \left(\frac{3 x}{x + 5} - 1\right)}{\left(x + 5\right)^{2}}$$

Simplify

3-я производная [src]

               /                                        2                          \
               |                            4*x      3*x                           |
               |                 x     1 - ----- + --------          x             |
               |          -1 + -----       5 + x          2   -1 + -----           |
  /       x  \ |    1          5 + x               (5 + x)         5 + x     2*x   |
4*|-1 + -----|*|- ----- - ---------- - -------------------- - ---------- - --------|
  \     5 + x/ |  5 + x       x                 x               5 + x             2|
               \                                                           (5 + x) /
------------------------------------------------------------------------------------
                                             2                                      
                                      (5 + x)

$$\frac{4 \left(\frac{x}{x + 5} - 1\right) \left(- \frac{2 x}{\left(x + 5\right)^{2}} - \frac{\frac{x}{x + 5} - 1}{x + 5} - \frac{1}{x + 5} - \frac{\frac{x}{x + 5} - 1}{x} - \frac{\frac{3 x^{2}}{\left(x + 5\right)^{2}} - \frac{4 x}{x + 5} + 1}{x}\right)}{\left(x + 5\right)^{2}}$$

Simplify

The third derivative [src]

               /                                        2                          \
               |                            4*x      3*x                           |
               |                 x     1 - ----- + --------          x             |
               |          -1 + -----       5 + x          2   -1 + -----           |
  /       x  \ |    1          5 + x               (5 + x)         5 + x     2*x   |
4*|-1 + -----|*|- ----- - ---------- - -------------------- - ---------- - --------|
  \     5 + x/ |  5 + x       x                 x               5 + x             2|
               \                                                           (5 + x) /
------------------------------------------------------------------------------------
                                             2                                      
                                      (5 + x)

$$\frac{4 \left(\frac{x}{x + 5} - 1\right) \left(- \frac{2 x}{\left(x + 5\right)^{2}} - \frac{\frac{x}{x + 5} - 1}{x + 5} - \frac{1}{x + 5} - \frac{\frac{x}{x + 5} - 1}{x} - \frac{\frac{3 x^{2}}{\left(x + 5\right)^{2}} - \frac{4 x}{x + 5} + 1}{x}\right)}{\left(x + 5\right)^{2}}$$

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (x/(x+5))^2

The graph:

Piecewise:

The solution