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e^(x)(sinx+cosx)
The derivative of the function/ e^(x)(sinx+cosx)

Derivative of e^(x)(sinx+cosx)

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

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The solution

You have entered [src] 
 x                  
e *(sin(x) + cos(x))
(sin⁡(x)+cos⁡(x))ex\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} 
d / x                  \
--\e *(sin(x) + cos(x))/
dx
ddx(sin⁡(x)+cos⁡(x))ex\frac{d}{d x} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}
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)=exf{\left(x \right)} = e^{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

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

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

    1. Differentiate sin⁡(x)+cos⁡(x)\sin{\left(x \right)} + \cos{\left(x \right)} term by term:

      1. The derivative of sine is cosine:

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

      2. The derivative of cosine is negative sine:

        ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

      The result is: −sin⁡(x)+cos⁡(x)- \sin{\left(x \right)} + \cos{\left(x \right)}

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

  2. Now simplify:

    2excos⁡(x)2 e^{x} \cos{\left(x \right)}

The answer is:

2excos⁡(x)2 e^{x} \cos{\left(x \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
(-sin(x) + cos(x))*e  + (sin(x) + cos(x))*e
(−sin⁡(x)+cos⁡(x))ex+(sin⁡(x)+cos⁡(x))ex\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{x}
Simplify
The second derivative [src] 
                       x
-2*(-cos(x) + sin(x))*e
−2(sin⁡(x)−cos⁡(x))ex- 2 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{x}
Simplify
The third derivative [src] 
    x       
-4*e *sin(x)
−4exsin⁡(x)- 4 e^{x} \sin{\left(x \right)}
Simplify
The graph
Derivative of e^(x)(sinx+cosx)
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