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The derivative of the function/ sqrt(8x+x^2)

Derivative of sqrt(8x+x^2)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   __________
  /        2 
\/  8*x + x
$$\sqrt{x^{2} + 8 x}$$ 
sqrt(8*x + x^2)
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      2. Apply the power rule: goes to

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    4 + x    
-------------
   __________
  /        2 
\/  8*x + x
$$\frac{x + 4}{\sqrt{x^{2} + 8 x}}$$
Simplify
The second derivative [src] 
            2
     (4 + x) 
1 - ---------
    x*(8 + x)
-------------
  ___________
\/ x*(8 + x)
$$\frac{1 - \frac{\left(x + 4\right)^{2}}{x \left(x + 8\right)}}{\sqrt{x \left(x + 8\right)}}$$
Simplify
The third derivative [src] 
  /             2\        
  |      (4 + x) |        
3*|-1 + ---------|*(4 + x)
  \     x*(8 + x)/        
--------------------------
                 3/2      
      (x*(8 + x))
$$\frac{3 \left(-1 + \frac{\left(x + 4\right)^{2}}{x \left(x + 8\right)}\right) \left(x + 4\right)}{\left(x \left(x + 8\right)\right)^{\frac{3}{2}}}$$
Simplify
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