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sinx*e^x
The derivative of the function/ sinx*e^x

Derivative of sinx*e^x

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

You have entered [src] 
        x
sin(x)*e
exsin⁡(x)e^{x} \sin{\left(x \right)} 
d /        x\
--\sin(x)*e /
dx
ddxexsin⁡(x)\frac{d}{d x} e^{x} \sin{\left(x \right)}
Transform
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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\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)}

    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: exsin⁡(x)+excos⁡(x)e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}

  2. Now simplify:

    2exsin⁡(x+π4)\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}

The answer is:

2exsin⁡(x+π4)\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}

The graph
02468-8-6-4-2-1010-5000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        x    x       
cos(x)*e  + e *sin(x)
exsin⁡(x)+excos⁡(x)e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}
Simplify
The second derivative [src] 
          x
2*cos(x)*e
2excos⁡(x)2 e^{x} \cos{\left(x \right)}
Simplify
The third derivative [src] 
                      x
2*(-sin(x) + cos(x))*e
2(−sin⁡(x)+cos⁡(x))ex2 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}
Simplify
The graph
Derivative of sinx*e^x
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