Mister Exam

Derivative of e^tan(x)

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

The graph:

from to

Piecewise:

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)}}
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:

      sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

    The result of the chain rule is:

    (sin2(x)+cos2(x))etan(x)cos2(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)cos2(x)\frac{e^{\tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}


The answer is:

etan(x)cos2(x)\frac{e^{\tan{\left(x \right)}}}{\cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-101002000000000000000
The first derivative [src]
/       2   \  tan(x)
\1 + tan (x)/*e      
(tan2(x)+1)etan(x)\left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}}
The second derivative [src]
/       2   \ /       2              \  tan(x)
\1 + tan (x)/*\1 + tan (x) + 2*tan(x)/*e      
(tan2(x)+1)(tan2(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)}}
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      
(tan2(x)+1)((tan2(x)+1)2+6(tan2(x)+1)tan(x)+6tan2(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)}}
The graph
Derivative of e^tan(x)