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Identical expressions
y=sqrt(x)/(x+ one)[ one -x)^ ten
y equally square root of (x) divide by (x plus 1)[1 minus x) to the power of 10
y equally square root of (x) divide by (x plus one)[ one minus x) to the power of ten
y=√(x)/(x+1)[1-x)^10
y=sqrt(x)/(x+1)[1-x)10
y=sqrtx/x+1[1-x10
y=sqrtx/x+1[1-x^10
y=sqrt(x) divide by (x+1)[1-x)^10
Similar expressions
y=sqrt(x)/(x-1)[1-x)^10
y=sqrt(x)/(x+1)[1+x)^10

The derivative of the function/ y=sqrt(x)/(x+1)[1-x)^10

Derivative of y=sqrt(x)/(x+1)[1-x)^10

The solution

You have entered [src]

  ___        10
\/ x *(1 - x)  
---------------
     x + 1

$$\frac{\sqrt{x} \left(1 - x\right)^{10}}{x + 1}$$

  /  ___        10\
d |\/ x *(1 - x)  |
--|---------------|
dx\     x + 1     /

$$\frac{d}{d x} \frac{\sqrt{x} \left(1 - x\right)^{10}}{x + 1}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{x} \left(1 - x\right)^{10}$ and $g{\left(x \right)} = x + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  $g{\left(x \right)} = \left(1 - x\right)^{10}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 1 - x$ .
  2. Apply the power rule: $u^{10}$ goes to $10 u^{9}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(1 - x\right)$ :
    1. Differentiate $1 - x$ 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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $-1$
      The result is: $-1$
    The result of the chain rule is:
    
    $- 10 \left(1 - x\right)^{9}$
  The result is: $- 10 \sqrt{x} \left(1 - x\right)^{9} + \frac{\left(1 - x\right)^{10}}{2 \sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

$\frac{- \sqrt{x} \left(1 - x\right)^{10} + \left(x + 1\right) \left(- 10 \sqrt{x} \left(1 - x\right)^{9} + \frac{\left(1 - x\right)^{10}}{2 \sqrt{x}}\right)}{\left(x + 1\right)^{2}}$
Now simplify:

$\frac{- x \left(x - 1\right)^{10} - \frac{\left(1 - x\right)^{9} \left(x + 1\right) \left(21 x - 1\right)}{2}}{\sqrt{x} \left(x + 1\right)^{2}}$

The answer is:

$\frac{- x \left(x - 1\right)^{10} - \frac{\left(1 - x\right)^{9} \left(x + 1\right) \left(21 x - 1\right)}{2}}{\sqrt{x} \left(x + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          10        ___        10        ___        9
   (1 - x)        \/ x *(1 - x)     10*\/ x *(1 - x) 
--------------- - --------------- - -----------------
    ___                      2            x + 1      
2*\/ x *(x + 1)       (x + 1)

$$- \frac{\sqrt{x} \left(1 - x\right)^{10}}{\left(x + 1\right)^{2}} - \frac{10 \sqrt{x} \left(1 - x\right)^{9}}{x + 1} + \frac{\left(1 - x\right)^{10}}{2 \sqrt{x} \left(x + 1\right)}$$

Simplify

The second derivative [src]

          /                                 2             2          ___                ___         2\
        8 |     ___   10*(-1 + x)   (-1 + x)      (-1 + x)      20*\/ x *(-1 + x)   2*\/ x *(-1 + x) |
(-1 + x) *|90*\/ x  + ----------- - --------- - ------------- - ----------------- + -----------------|
          |                ___           3/2      ___                 1 + x                     2    |
          \              \/ x         4*x       \/ x *(1 + x)                            (1 + x)     /
------------------------------------------------------------------------------------------------------
                                                1 + x

$$\frac{\left(x - 1\right)^{8} \cdot \left(\frac{2 \sqrt{x} \left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} - \frac{20 \sqrt{x} \left(x - 1\right)}{x + 1} + 90 \sqrt{x} - \frac{\left(x - 1\right)^{2}}{\sqrt{x} \left(x + 1\right)} + \frac{10 \left(x - 1\right)}{\sqrt{x}} - \frac{\left(x - 1\right)^{2}}{4 x^{\frac{3}{2}}}\right)}{x + 1}$$

Simplify

The third derivative [src]

            /                                    2           3             3           ___                        2       ___         3        ___         2             3   \
          7 |      ___   45*(-1 + x)   5*(-1 + x)    (-1 + x)      (-1 + x)       90*\/ x *(-1 + x)    10*(-1 + x)    2*\/ x *(-1 + x)    20*\/ x *(-1 + x)      (-1 + x)    |
3*(-1 + x) *|240*\/ x  + ----------- - ----------- + --------- + -------------- - ----------------- - ------------- - ----------------- + ------------------ + --------------|
            |                 ___            3/2          5/2      ___        2         1 + x           ___                       3                   2           3/2        |
            \               \/ x          2*x          8*x       \/ x *(1 + x)                        \/ x *(1 + x)        (1 + x)             (1 + x)         4*x   *(1 + x)/
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    1 + x

$$\frac{3 \left(x - 1\right)^{7} \left(- \frac{2 \sqrt{x} \left(x - 1\right)^{3}}{\left(x + 1\right)^{3}} + \frac{20 \sqrt{x} \left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} - \frac{90 \sqrt{x} \left(x - 1\right)}{x + 1} + 240 \sqrt{x} + \frac{\left(x - 1\right)^{3}}{\sqrt{x} \left(x + 1\right)^{2}} - \frac{10 \left(x - 1\right)^{2}}{\sqrt{x} \left(x + 1\right)} + \frac{45 \left(x - 1\right)}{\sqrt{x}} + \frac{\left(x - 1\right)^{3}}{4 x^{\frac{3}{2}} \left(x + 1\right)} - \frac{5 \left(x - 1\right)^{2}}{2 x^{\frac{3}{2}}} + \frac{\left(x - 1\right)^{3}}{8 x^{\frac{5}{2}}}\right)}{x + 1}$$

Simplify

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Derivative of y=sqrt(x)/(x+1)[1-x)^10

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