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      
$$e^{\tan{\left(x \right)}}$$
d / tan(x)\
--\e      /
dx         
$$\frac{d}{d x} e^{\tan{\left(x \right)}}$$
Detail solution
  1. Let .

  2. The derivative of is itself.

  3. Then, apply the chain rule. Multiply by :

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. The derivative of sine is cosine:

      To find :

      1. The derivative of cosine is negative sine:

      Now plug in to the quotient rule:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
/       2   \  tan(x)
\1 + tan (x)/*e      
$$\left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}}$$
The second derivative [src]
/       2   \ /       2              \  tan(x)
\1 + tan (x)/*\1 + tan (x) + 2*tan(x)/*e      
$$\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}}$$
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      
$$\left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 6 \tan^{2}{\left(x \right)} + 2\right) e^{\tan{\left(x \right)}}$$
The graph
Derivative of e^tan(x)