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The derivative of the function/ sqrtx^3-sqrtx+2

Derivative of sqrtx^3-sqrtx+2

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

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The solution

You have entered [src] 
     3            
  ___      ___    
\/ x   - \/ x  + 2
((x)3−x)+2\left(\left(\sqrt{x}\right)^{3} - \sqrt{x}\right) + 2 
(sqrt(x))^3 - sqrt(x) + 2
Detail solution

  1. Differentiate ((x)3−x)+2\left(\left(\sqrt{x}\right)^{3} - \sqrt{x}\right) + 2 term by term:

    1. Differentiate (x)3−x\left(\sqrt{x}\right)^{3} - \sqrt{x} term by term:

      1. Let u=xu = \sqrt{x}.

      2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

      3. Then, apply the chain rule. Multiply by ddxx\frac{d}{d x} \sqrt{x}:

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

        The result of the chain rule is:

        3x2\frac{3 \sqrt{x}}{2}

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

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

        So, the result is: −12x- \frac{1}{2 \sqrt{x}}

      The result is: 3x2−12x\frac{3 \sqrt{x}}{2} - \frac{1}{2 \sqrt{x}}

    2. The derivative of the constant 22 is zero.

    The result is: 3x2−12x\frac{3 \sqrt{x}}{2} - \frac{1}{2 \sqrt{x}}

  2. Now simplify:

    3x−12x\frac{3 x - 1}{2 \sqrt{x}}

The answer is:

3x−12x\frac{3 x - 1}{2 \sqrt{x}}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
               3/2
     1      3*x   
- ------- + ------
      ___    2*x  
  2*\/ x
3x322x−12x\frac{3 x^{\frac{3}{2}}}{2 x} - \frac{1}{2 \sqrt{x}}
Simplify
The second derivative [src] 
     1 
 3 + - 
     x 
-------
    ___
4*\/ x
3+1x4x\frac{3 + \frac{1}{x}}{4 \sqrt{x}}
Simplify
The third derivative [src] 
   /    1\
-3*|1 + -|
   \    x/
----------
     3/2  
  8*x
−3(1+1x)8x32- \frac{3 \left(1 + \frac{1}{x}\right)}{8 x^{\frac{3}{2}}}
Simplify
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