Mister Exam

Derivative of x*e^(2x)

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

from to

Piecewise:

The solution

You have entered [src]
   2*x
x*e   
xe2xx e^{2 x}
d /   2*x\
--\x*e   /
dx        
ddxxe2x\frac{d}{d x} x 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)=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}

  2. Now simplify:

    (2x+1)e2x\left(2 x + 1\right) e^{2 x}


The answer is:

(2x+1)e2x\left(2 x + 1\right) e^{2 x}

The graph
02468-8-6-4-2-101020000000000-10000000000
The first derivative [src]
 2*x        2*x
e    + 2*x*e   
2xe2x+e2x2 x e^{2 x} + e^{2 x}
The second derivative [src]
           2*x
4*(1 + x)*e   
4(x+1)e2x4 \left(x + 1\right) e^{2 x}
The third derivative [src]
             2*x
4*(3 + 2*x)*e   
4(2x+3)e2x4 \cdot \left(2 x + 3\right) e^{2 x}
The graph
Derivative of x*e^(2x)