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y=x*exp(x)*sin(x)
The derivative of the function/ y=x*exp(x)*sin(x)

Derivative of y=x*exp(x)*sin(x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   x       
x*e *sin(x)
xexsin⁡(x)x e^{x} \sin{\left(x \right)} 
(x*exp(x))*sin(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)=xexf{\left(x \right)} = x e^{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

      1. The derivative of exe^{x} is itself.

      The result is: xex+exx e^{x} + e^{x}

    g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result is: xexcos⁡(x)+(xex+ex)sin⁡(x)x e^{x} \cos{\left(x \right)} + \left(x e^{x} + e^{x}\right) \sin{\left(x \right)}

  2. Now simplify:

    (xcos⁡(x)+(x+1)sin⁡(x))ex\left(x \cos{\left(x \right)} + \left(x + 1\right) \sin{\left(x \right)}\right) e^{x}

The answer is:

(xcos⁡(x)+(x+1)sin⁡(x))ex\left(x \cos{\left(x \right)} + \left(x + 1\right) \sin{\left(x \right)}\right) e^{x}

The graph
02468-8-6-4-2-1010-500000500000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/   x    x\                    x
\x*e  + e /*sin(x) + x*cos(x)*e
xexcos⁡(x)+(xex+ex)sin⁡(x)x e^{x} \cos{\left(x \right)} + \left(x e^{x} + e^{x}\right) \sin{\left(x \right)}
Simplify
The second derivative [src] 
                                                x
((2 + x)*sin(x) - x*sin(x) + 2*(1 + x)*cos(x))*e
(−xsin⁡(x)+2(x+1)cos⁡(x)+(x+2)sin⁡(x))ex\left(- x \sin{\left(x \right)} + 2 \left(x + 1\right) \cos{\left(x \right)} + \left(x + 2\right) \sin{\left(x \right)}\right) e^{x}
Simplify
The third derivative [src] 
                                                                   x
((3 + x)*sin(x) - x*cos(x) - 3*(1 + x)*sin(x) + 3*(2 + x)*cos(x))*e
(−xcos⁡(x)−3(x+1)sin⁡(x)+3(x+2)cos⁡(x)+(x+3)sin⁡(x))ex\left(- x \cos{\left(x \right)} - 3 \left(x + 1\right) \sin{\left(x \right)} + 3 \left(x + 2\right) \cos{\left(x \right)} + \left(x + 3\right) \sin{\left(x \right)}\right) e^{x}
Simplify
The graph
Derivative of y=x*exp(x)*sin(x)
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