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

Derivative of (x+4)/(sqrt(x))

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
x + 4
-----
  ___
\/ x
x+4x\frac{x + 4}{\sqrt{x}} 
d /x + 4\
--|-----|
dx|  ___|
  \\/ x /
ddxx+4x\frac{d}{d x} \frac{x + 4}{\sqrt{x}}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x+4f{\left(x \right)} = x + 4 and g(x)=xg{\left(x \right)} = \sqrt{x}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate x+4x + 4 term by term:

      1. The derivative of the constant 44 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

    Now plug in to the quotient rule:

    x−x+42xx\frac{\sqrt{x} - \frac{x + 4}{2 \sqrt{x}}}{x}

  2. Now simplify:

    x−42x32\frac{x - 4}{2 x^{\frac{3}{2}}}

The answer is:

x−42x32\frac{x - 4}{2 x^{\frac{3}{2}}}

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  1     x + 4 
----- - ------
  ___      3/2
\/ x    2*x
1x−x+42x32\frac{1}{\sqrt{x}} - \frac{x + 4}{2 x^{\frac{3}{2}}}
Simplify
The second derivative [src] 
     3*(4 + x)
-1 + ---------
        4*x   
--------------
      3/2     
     x
−1+3(x+4)4xx32\frac{-1 + \frac{3 \left(x + 4\right)}{4 x}}{x^{\frac{3}{2}}}
Simplify
The third derivative [src] 
  /    5*(4 + x)\
3*|6 - ---------|
  \        x    /
-----------------
         5/2     
      8*x
3⋅(6−5(x+4)x)8x52\frac{3 \cdot \left(6 - \frac{5 \left(x + 4\right)}{x}\right)}{8 x^{\frac{5}{2}}}
Simplify
The graph
Derivative of (x+4)/(sqrt(x))
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