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sqrt(x^2+3x)
The derivative of the function/ sqrt(x^2+3x)

Derivative of sqrt(x^2+3x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   __________
  /  2       
\/  x  + 3*x
$$\sqrt{x^{2} + 3 x}$$ 
  /   __________\
d |  /  2       |
--\\/  x  + 3*x /
dx
$$\frac{d}{d x} \sqrt{x^{2} + 3 x}$$
Transform
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. Apply the power rule: goes to

      2. 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:

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