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y=x(sqrtx^3)(sqrtx)
The derivative of the function/ y=x(sqrtx^3)(sqrtx)

Derivative of y=x(sqrtx^3)(sqrtx)

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

You have entered [src] 
       3      
    ___    ___
x*\/ x  *\/ x
xx(x)3\sqrt{x} x \left(\sqrt{x}\right)^{3} 
  /       3      \
d |    ___    ___|
--\x*\/ x  *\/ x /
dx
ddxxx(x)3\frac{d}{d x} \sqrt{x} x \left(\sqrt{x}\right)^{3}
Transform
Detail solution

  1. Apply the product rule:

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

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

    1. Apply the power rule: xx goes to 11

    g(x)=(x)3g{\left(x \right)} = \left(\sqrt{x}\right)^{3}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    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}

    h(x)=xh{\left(x \right)} = \sqrt{x}; to find ddxh(x)\frac{d}{d x} h{\left(x \right)}:

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

    The result is: 3xx322+3x22\frac{3 \sqrt{x} x^{\frac{3}{2}}}{2} + \frac{3 x^{2}}{2}

  2. Now simplify:

    3x23 x^{2}

The answer is:

3x23 x^{2}

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   2       ___  3/2
3*x    3*\/ x *x   
---- + ------------
 2          2
3xx322+3x22\frac{3 \sqrt{x} x^{\frac{3}{2}}}{2} + \frac{3 x^{2}}{2}
Simplify
The second derivative [src] 
6*x
6x6 x
Simplify
The third derivative [src] 
6
66
Simplify
The graph
Derivative of y=x(sqrtx^3)(sqrtx)
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