Mister Exam

Derivative of xe^(2x-1)

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 - 1
x*e       
xe2x1x e^{2 x - 1}
d /   2*x - 1\
--\x*e       /
dx            
ddxxe2x1\frac{d}{d x} x e^{2 x - 1}
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)=e2x1g{\left(x \right)} = e^{2 x - 1}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=2x1u = 2 x - 1.

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

    3. Then, apply the chain rule. Multiply by ddx(2x1)\frac{d}{d x} \left(2 x - 1\right):

      1. Differentiate 2x12 x - 1 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: xx goes to 11

          So, the result is: 22

        2. The derivative of the constant (1)1\left(-1\right) 1 is zero.

        The result is: 22

      The result of the chain rule is:

      2e2x12 e^{2 x - 1}

    The result is: 2xe2x1+e2x12 x e^{2 x - 1} + e^{2 x - 1}

  2. Now simplify:

    (2x+1)e2x1\left(2 x + 1\right) e^{2 x - 1}


The answer is:

(2x+1)e2x1\left(2 x + 1\right) e^{2 x - 1}

The graph
02468-8-6-4-2-1010-50000000005000000000
The first derivative [src]
 2*x - 1        2*x - 1
e        + 2*x*e       
2xe2x1+e2x12 x e^{2 x - 1} + e^{2 x - 1}
The second derivative [src]
           -1 + 2*x
4*(1 + x)*e        
4(x+1)e2x14 \left(x + 1\right) e^{2 x - 1}
The third derivative [src]
             -1 + 2*x
4*(3 + 2*x)*e        
4(2x+3)e2x14 \cdot \left(2 x + 3\right) e^{2 x - 1}
The graph
Derivative of xe^(2x-1)