Mister Exam

Derivative of exp(x)*tan(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 x       
e *tan(x)
$$e^{x} \tan{\left(x \right)}$$
exp(x)*tan(x)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. The derivative of is itself.

    ; to find :

    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 is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
/       2   \  x    x       
\1 + tan (x)/*e  + e *tan(x)
$$\left(\tan^{2}{\left(x \right)} + 1\right) e^{x} + e^{x} \tan{\left(x \right)}$$
The second derivative [src]
/         2        /       2   \                \  x
\2 + 2*tan (x) + 2*\1 + tan (x)/*tan(x) + tan(x)/*e 
$$\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) e^{x}$$
The third derivative [src]
/         2        /       2   \ /         2   \     /       2   \                \  x
\3 + 3*tan (x) + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + 6*\1 + tan (x)/*tan(x) + tan(x)/*e 
$$\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 3\right) e^{x}$$
The graph
Derivative of exp(x)*tan(x)