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Identical expressions
sqrt10(x/(x+ five))
square root of 10(x divide by (x plus 5))
square root of 10(x divide by (x plus five))
√10(x/(x+5))
sqrt10x/x+5
sqrt10(x divide by (x+5))
Similar expressions
sqrt10(x/(x-5))

The derivative of the function/ sqrt10(x/(x+5))

Derivative of sqrt10(x/(x+5))

The solution

You have entered [src]

       0.1
/  x  \   
|-----|   
\x + 5/

$$\left(\frac{x}{x + 5}\right)^{0.1}$$

(x/(x + 5))^0.1

Detail solution

Let $u = \frac{x}{x + 5}$ .
Apply the power rule: $u^{0.1}$ goes to $\frac{0.1}{u^{0.9}}$
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:

$\frac{0.5}{\left(\frac{x}{x + 5}\right)^{0.9} \left(x + 5\right)^{2}}$
Now simplify:

$\frac{0.5}{\left(\frac{x}{x + 5}\right)^{0.9} \left(x + 5\right)^{2}}$

The answer is:

$\frac{0.5}{\left(\frac{x}{x + 5}\right)^{0.9} \left(x + 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       0.1                           
/  x  \            / 0.1     0.1*x  \
|-----|   *(x + 5)*|----- - --------|
\x + 5/            |x + 5          2|
                   \        (x + 5) /
-------------------------------------
                  x

$$\frac{\left(\frac{x}{x + 5}\right)^{0.1} \left(x + 5\right) \left(- \frac{0.1 x}{\left(x + 5\right)^{2}} + \frac{0.1}{x + 5}\right)}{x}$$

Simplify

The second derivative [src]

                        /                      0.01*x\
       0.1              |              -0.01 + ------|
/  x  \    /       x  \ |0.1    0.1            5 + x |
|-----|   *|-1 + -----|*|--- + ----- + --------------|
\5 + x/    \     5 + x/ \ x    5 + x         x       /
------------------------------------------------------
                          x

$$\frac{\left(\frac{x}{x + 5}\right)^{0.1} \left(\frac{x}{x + 5} - 1\right) \left(\frac{0.1}{x + 5} + \frac{\frac{0.01 x}{x + 5} - 0.01}{x} + \frac{0.1}{x}\right)}{x}$$

Simplify

The third derivative [src]

                        /                                                                  2                                                   \
                        |                           0.03*x                     /       x  \            0.03*x           0.02*x           0.04*x|
       0.1              |                   -0.03 + ------               0.001*|-1 + -----|    -0.03 + ------   -0.02 + ------   -0.04 + ------|
/  x  \    /       x  \ |    0.2      0.2           5 + x       0.2            \     5 + x/            5 + x            5 + x            5 + x |
|-----|   *|-1 + -----|*|- -------- - --- - -------------- - --------- - ------------------- + -------------- - -------------- - --------------|
\5 + x/    \     5 + x/ |         2     2          2         x*(5 + x)             2             x*(5 + x)        x*(5 + x)        x*(5 + x)   |
                        \  (5 + x)     x          x                               x                                                            /
------------------------------------------------------------------------------------------------------------------------------------------------
                                                                       x

$$\frac{\left(\frac{x}{x + 5}\right)^{0.1} \left(\frac{x}{x + 5} - 1\right) \left(- \frac{0.2}{\left(x + 5\right)^{2}} - \frac{\frac{0.02 x}{x + 5} - 0.02}{x \left(x + 5\right)} + \frac{\frac{0.03 x}{x + 5} - 0.03}{x \left(x + 5\right)} - \frac{\frac{0.04 x}{x + 5} - 0.04}{x \left(x + 5\right)} - \frac{0.2}{x \left(x + 5\right)} - \frac{\frac{0.03 x}{x + 5} - 0.03}{x^{2}} - \frac{0.001 \left(\frac{x}{x + 5} - 1\right)^{2}}{x^{2}} - \frac{0.2}{x^{2}}\right)}{x}$$

Simplify

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Derivative of sqrt10(x/(x+5))

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Piecewise:

The solution