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The derivative of the function/ (e^x*sin(x)*tg(x))

Derivative of (e^x*sin(x)*tg(x))

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

You have entered [src] 
 x              
E *sin(x)*tan(x)
exsin⁡(x)tan⁡(x)e^{x} \sin{\left(x \right)} \tan{\left(x \right)} 
(E^x*sin(x))*tan(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)=exsin⁡(x)f{\left(x \right)} = e^{x} \sin{\left(x \right)}; 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)=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)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: exsin⁡(x)+excos⁡(x)e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}

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

    1. Rewrite the function to be differentiated:

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

    2. Apply the quotient rule, which is:

      ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

      f(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\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)}

      To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. The derivative of cosine is negative sine:

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

      Now plug in to the quotient rule:

      sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result is: (exsin⁡(x)+excos⁡(x))tan⁡(x)+(sin⁡2(x)+cos⁡2(x))exsin⁡(x)cos⁡2(x)\left(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}\right) \tan{\left(x \right)} + \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

  2. Now simplify:

    (2sin⁡(x+π4)cos⁡(x)+1)exsin⁡(x)cos⁡2(x)\frac{\left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(x \right)} + 1\right) e^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

The answer is:

(2sin⁡(x+π4)cos⁡(x)+1)exsin⁡(x)cos⁡2(x)\frac{\left(\sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(x \right)} + 1\right) e^{x} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}

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       \          /       2   \  x       
\cos(x)*e  + e *sin(x)/*tan(x) + \1 + tan (x)/*e *sin(x)
(exsin⁡(x)+excos⁡(x))tan⁡(x)+(tan⁡2(x)+1)exsin⁡(x)\left(e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)}\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) e^{x} \sin{\left(x \right)}
Simplify
The second derivative [src] 
  //       2   \                                     /       2   \              \  x
2*\\1 + tan (x)/*(cos(x) + sin(x)) + cos(x)*tan(x) + \1 + tan (x)/*sin(x)*tan(x)/*e
2((sin⁡(x)+cos⁡(x))(tan⁡2(x)+1)+(tan⁡2(x)+1)sin⁡(x)tan⁡(x)+cos⁡(x)tan⁡(x))ex2 \left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)}\right) e^{x}
Simplify
The third derivative [src] 
  /                               /       2   \          /       2   \ /         2   \            /       2   \                         \  x
2*\-(-cos(x) + sin(x))*tan(x) + 3*\1 + tan (x)/*cos(x) + \1 + tan (x)/*\1 + 3*tan (x)/*sin(x) + 3*\1 + tan (x)/*(cos(x) + sin(x))*tan(x)/*e
2(−(sin⁡(x)−cos⁡(x))tan⁡(x)+3(sin⁡(x)+cos⁡(x))(tan⁡2(x)+1)tan⁡(x)+(tan⁡2(x)+1)(3tan⁡2(x)+1)sin⁡(x)+3(tan⁡2(x)+1)cos⁡(x))ex2 \left(- \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \tan{\left(x \right)} + 3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)}\right) e^{x}
Simplify
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