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sqrtx*(3x^2-4x)
The derivative of the function/ sqrtx*(3x^2-4x)

Derivative of sqrtx*(3x^2-4x)

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

You have entered [src] 
  ___ /   2      \
\/ x *\3*x  - 4*x/
x(3x2−4x)\sqrt{x} \left(3 x^{2} - 4 x\right) 
d /  ___ /   2      \\
--\\/ x *\3*x  - 4*x//
dx
ddxx(3x2−4x)\frac{d}{d x} \sqrt{x} \left(3 x^{2} - 4 x\right)
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=xf{\left(x \right)} = \sqrt{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

    g(x)=3x2−4xg{\left(x \right)} = 3 x^{2} - 4 x; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate 3x2−4x3 x^{2} - 4 x 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: x2x^{2} goes to 2x2 x

        So, the result is: 6x6 x

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

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

          1. Apply the power rule: xx goes to 11

          So, the result is: 44

        So, the result is: −4-4

      The result is: 6x−46 x - 4

    The result is: x(6x−4)+3x2−4x2x\sqrt{x} \left(6 x - 4\right) + \frac{3 x^{2} - 4 x}{2 \sqrt{x}}

  2. Now simplify:

    x(15x2−6)\sqrt{x} \left(\frac{15 x}{2} - 6\right)

The answer is:

x(15x2−6)\sqrt{x} \left(\frac{15 x}{2} - 6\right)

The graph
02468-8-6-4-2-1010-10001000
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Plot the graph f'(x)
The first derivative [src] 
                      2      
  ___              3*x  - 4*x
\/ x *(-4 + 6*x) + ----------
                        ___  
                    2*\/ x
x(6x−4)+3x2−4x2x\sqrt{x} \left(6 x - 4\right) + \frac{3 x^{2} - 4 x}{2 \sqrt{x}}
Simplify
The second derivative [src] 
    ___   2*(-2 + 3*x)   -4 + 3*x
6*\/ x  + ------------ - --------
               ___           ___ 
             \/ x        4*\/ x
6x−3x−44x+2⋅(3x−2)x6 \sqrt{x} - \frac{3 x - 4}{4 \sqrt{x}} + \frac{2 \cdot \left(3 x - 2\right)}{\sqrt{x}}
Simplify
The third derivative [src] 
  /    -2 + 3*x   -4 + 3*x\
3*|3 - -------- + --------|
  \      2*x        8*x   /
---------------------------
             ___           
           \/ x
3⋅(3+3x−48x−3x−22x)x\frac{3 \cdot \left(3 + \frac{3 x - 4}{8 x} - \frac{3 x - 2}{2 x}\right)}{\sqrt{x}}
Simplify
The graph
Derivative of sqrtx*(3x^2-4x)
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