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

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

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

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

    1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

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

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\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 of the chain rule is:

      esin⁡(x)cos⁡(x)e^{\sin{\left(x \right)}} \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: (sin⁡2(x)+cos⁡2(x))esin⁡(x)cos⁡2(x)+esin⁡(x)cos⁡(x)tan⁡(x)\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{\sin{\left(x \right)}}}{\cos^{2}{\left(x \right)}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} \tan{\left(x \right)}

  2. Now simplify:

    (−sin⁡3(x)+sin⁡(x)+1)esin⁡(x)cos⁡2(x)\frac{\left(- \sin^{3}{\left(x \right)} + \sin{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

The answer is:

(−sin⁡3(x)+sin⁡(x)+1)esin⁡(x)cos⁡2(x)\frac{\left(- \sin^{3}{\left(x \right)} + \sin{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-25002500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/       2   \  sin(x)           sin(x)       
\1 + tan (x)/*e       + cos(x)*e      *tan(x)
(tan⁡2(x)+1)esin⁡(x)+esin⁡(x)cos⁡(x)tan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) e^{\sin{\left(x \right)}} + e^{\sin{\left(x \right)}} \cos{\left(x \right)} \tan{\left(x \right)}
Simplify
The second derivative [src] 
/  /     2            \            /       2   \            /       2   \       \  sin(x)
\- \- cos (x) + sin(x)/*tan(x) + 2*\1 + tan (x)/*cos(x) + 2*\1 + tan (x)/*tan(x)/*e
(−(sin⁡(x)−cos⁡2(x))tan⁡(x)+2(tan⁡2(x)+1)cos⁡(x)+2(tan⁡2(x)+1)tan⁡(x))esin⁡(x)\left(- \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) e^{\sin{\left(x \right)}}
Simplify
The third derivative [src] 
/    /       2   \ /     2            \     /       2   \ /         2   \   /       2              \                   /       2   \              \  sin(x)
\- 3*\1 + tan (x)/*\- cos (x) + sin(x)/ + 2*\1 + tan (x)/*\1 + 3*tan (x)/ - \1 - cos (x) + 3*sin(x)/*cos(x)*tan(x) + 6*\1 + tan (x)/*cos(x)*tan(x)/*e
(−3(sin⁡(x)−cos⁡2(x))(tan⁡2(x)+1)+2(tan⁡2(x)+1)(3tan⁡2(x)+1)+6(tan⁡2(x)+1)cos⁡(x)tan⁡(x)−(3sin⁡(x)−cos⁡2(x)+1)cos⁡(x)tan⁡(x))esin⁡(x)\left(- 3 \left(\sin{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - \left(3 \sin{\left(x \right)} - \cos^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)}\right) e^{\sin{\left(x \right)}}
Simplify
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