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

Derivative of e^tan(x)

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

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

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

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

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

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

  4. Now simplify:

    etan⁡(x)cos⁡2(x)\frac{e^{\tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

The answer is:

etan⁡(x)cos⁡2(x)\frac{e^{\tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-101002000000000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
/       2   \  tan(x)
\1 + tan (x)/*e
(tan⁡2(x)+1)etan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}}
Simplify
The second derivative [src] 
/       2   \ /       2              \  tan(x)
\1 + tan (x)/*\1 + tan (x) + 2*tan(x)/*e
(tan⁡2(x)+1)(tan⁡2(x)+2tan⁡(x)+1)etan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}}
Simplify
The third derivative [src] 
              /                 2                                     \        
/       2   \ |    /       2   \         2        /       2   \       |  tan(x)
\1 + tan (x)/*\2 + \1 + tan (x)/  + 6*tan (x) + 6*\1 + tan (x)/*tan(x)/*e
(tan⁡2(x)+1)((tan⁡2(x)+1)2+6(tan⁡2(x)+1)tan⁡(x)+6tan⁡2(x)+2)etan⁡(x)\left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 6 \tan^{2}{\left(x \right)} + 2\right) e^{\tan{\left(x \right)}}
Simplify
The graph
Derivative of e^tan(x)
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