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tan(x)/e^x
The derivative of the function/ tan(x)/e^x

Derivative of tan(x)/e^x

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

You have entered [src] 
tan(x)
------
   x  
  e
tan⁡(x)ex\frac{\tan{\left(x \right)}}{e^{x}} 
d /tan(x)\
--|------|
dx|   x  |
  \  e   /
ddxtan⁡(x)ex\frac{d}{d x} \frac{\tan{\left(x \right)}}{e^{x}}
Transform
Detail solution

  1. 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)=tan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} and g(x)=exg{\left(x \right)} = e^{x}.

    To find ddxf(x)\frac{d}{d x} f{\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)}}

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

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

    Now plug in to the quotient rule:

    ((sin⁡2(x)+cos⁡2(x))excos⁡2(x)−extan⁡(x))e−2x\left(\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x}}{\cos^{2}{\left(x \right)}} - e^{x} \tan{\left(x \right)}\right) e^{- 2 x}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-500000500000
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The first derivative [src] 
/       2   \  -x    -x       
\1 + tan (x)/*e   - e  *tan(x)
(tan⁡2(x)+1)e−x−e−xtan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) e^{- x} - e^{- x} \tan{\left(x \right)}
Simplify
The second derivative [src] 
/          2        /       2   \                \  -x
\-2 - 2*tan (x) + 2*\1 + tan (x)/*tan(x) + tan(x)/*e
(2(tan⁡2(x)+1)tan⁡(x)−2tan⁡2(x)+tan⁡(x)−2)e−x\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - 2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} - 2\right) e^{- x}
Simplify
The third derivative [src] 
/                  2        /       2   \            /       2   \ /         2   \\  -x
\3 - tan(x) + 3*tan (x) - 6*\1 + tan (x)/*tan(x) + 2*\1 + tan (x)/*\1 + 3*tan (x)//*e
(2(tan⁡2(x)+1)(3tan⁡2(x)+1)−6(tan⁡2(x)+1)tan⁡(x)+3tan⁡2(x)−tan⁡(x)+3)e−x\left(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) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} - \tan{\left(x \right)} + 3\right) e^{- x}
Simplify
The graph
Derivative of tan(x)/e^x
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