Mister Exam

Derivative of tan(x)*sin(x)

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

The graph:

from to

Piecewise:

The solution

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

    ; 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:

    ; to find :

    1. The derivative of sine is cosine:

    The result is:

  2. Now simplify:


The answer is:

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