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x^2*e^(2*x)

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x^2*e^(2*x)

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Derivative of x^2*e^(2*x)

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

You have entered [src]
 2  2*x
x *e   
x2e2xx^{2} e^{2 x}
d / 2  2*x\
--\x *e   /
dx         
ddxx2e2x\frac{d}{d x} x^{2} e^{2 x}
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)=x2f{\left(x \right)} = x^{2}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x2x^{2} goes to 2x2 x

    g(x)=e2xg{\left(x \right)} = e^{2 x}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=2xu = 2 x.

    2. The derivative of eue^{u} is itself.

    3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

      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: 22

      The result of the chain rule is:

      2e2x2 e^{2 x}

    The result is: 2x2e2x+2xe2x2 x^{2} e^{2 x} + 2 x e^{2 x}

  2. Now simplify:

    2x(x+1)e2x2 x \left(x + 1\right) e^{2 x}


The answer is:

2x(x+1)e2x2 x \left(x + 1\right) e^{2 x}

The graph
02468-8-6-4-2-1010200000000000-100000000000
The first derivative [src]
     2*x      2  2*x
2*x*e    + 2*x *e   
2x2e2x+2xe2x2 x^{2} e^{2 x} + 2 x e^{2 x}
The second derivative [src]
  /       2      \  2*x
2*\1 + 2*x  + 4*x/*e   
2(2x2+4x+1)e2x2 \cdot \left(2 x^{2} + 4 x + 1\right) e^{2 x}
The third derivative [src]
  /       2      \  2*x
4*\3 + 2*x  + 6*x/*e   
4(2x2+6x+3)e2x4 \cdot \left(2 x^{2} + 6 x + 3\right) e^{2 x}
The graph
Derivative of x^2*e^(2*x)