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1-e^(2x)-2xe^(2x)

Derivative of 1-e^(2x)-2xe^(2x)

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

You have entered [src]
     2*x        2*x
1 - e    - 2*x*e   
2xe2xe2x+1- 2 x e^{2 x} - e^{2 x} + 1
d /     2*x        2*x\
--\1 - e    - 2*x*e   /
dx                     
ddx(2xe2xe2x+1)\frac{d}{d x} \left(- 2 x e^{2 x} - e^{2 x} + 1\right)
Detail solution
  1. Differentiate 2xe2xe2x+1- 2 x e^{2 x} - e^{2 x} + 1 term by term:

    1. The derivative of the constant 11 is zero.

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

      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}

      So, the result is: 2e2x- 2 e^{2 x}

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

          1. Apply the power rule: xx goes to 11

          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: 2xe2x+e2x2 x e^{2 x} + e^{2 x}

        So, the result is: 4xe2x+2e2x4 x e^{2 x} + 2 e^{2 x}

      So, the result is: 4xe2x2e2x- 4 x e^{2 x} - 2 e^{2 x}

    The result is: 4xe2x4e2x- 4 x e^{2 x} - 4 e^{2 x}

  2. Now simplify:

    4(x1)e2x4 \left(- x - 1\right) e^{2 x}


The answer is:

4(x1)e2x4 \left(- x - 1\right) e^{2 x}

The graph
02468-8-6-4-2-1010-2500000000025000000000
The first derivative [src]
     2*x        2*x
- 4*e    - 4*x*e   
4xe2x4e2x- 4 x e^{2 x} - 4 e^{2 x}
The second derivative [src]
              2*x
-4*(3 + 2*x)*e   
4(2x+3)e2x- 4 \cdot \left(2 x + 3\right) e^{2 x}
The third derivative [src]
             2*x
-16*(2 + x)*e   
16(x+2)e2x- 16 \left(x + 2\right) e^{2 x}
The graph
Derivative of 1-e^(2x)-2xe^(2x)