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Derivative of:
Derivative of 4/x^2
Derivative of cos(x^3)
Derivative of (cos(x))^3
Derivative of 1/tan(x)
Identical expressions
sqrt(x)*(- four)/(x+ two)
square root of (x) multiply by ( minus 4) divide by (x plus 2)
square root of (x) multiply by ( minus four) divide by (x plus two)
√(x)*(-4)/(x+2)
sqrt(x)(-4)/(x+2)
sqrtx-4/x+2
sqrt(x)*(-4) divide by (x+2)
Similar expressions
sqrt(x)*(4)/(x+2)
sqrt(x)*(-4)/(x-2)

The derivative of the function/ sqrt(x)*(-4)/(x+2)

Derivative of sqrt(x)*(-4)/(x+2)

The solution

You have entered [src]

  ___     
\/ x *(-4)
----------
  x + 2

$$\frac{\left(-4\right) \sqrt{x}}{x + 2}$$

(sqrt(x)*(-4))/(x + 2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - 4 \sqrt{x}$ and $g{\left(x \right)} = x + 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  So, the result is: $- \frac{2}{\sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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: $1$
Now plug in to the quotient rule:

$\frac{4 \sqrt{x} - \frac{2 \left(x + 2\right)}{\sqrt{x}}}{\left(x + 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      ___ 
        2         4*\/ x  
- ------------- + --------
    ___                  2
  \/ x *(x + 2)   (x + 2)

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

Simplify

The second derivative [src]

           ___                 
 1     8*\/ x           4      
---- - -------- + -------------
 3/2          2     ___        
x      (2 + x)    \/ x *(2 + x)
-------------------------------
             2 + x

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

Simplify

The third derivative [src]

  /                                               ___ \
  |    1           1               4          8*\/ x  |
3*|- ------ - ------------ - -------------- + --------|
  |     5/2    3/2             ___        2          3|
  \  2*x      x   *(2 + x)   \/ x *(2 + x)    (2 + x) /
-------------------------------------------------------
                         2 + x

$$\frac{3 \left(\frac{8 \sqrt{x}}{\left(x + 2\right)^{3}} - \frac{4}{\sqrt{x} \left(x + 2\right)^{2}} - \frac{1}{x^{\frac{3}{2}} \left(x + 2\right)} - \frac{1}{2 x^{\frac{5}{2}}}\right)}{x + 2}$$

Simplify

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Derivative of sqrt(x)*(-4)/(x+2)

The graph:

Piecewise:

The solution