Mister Exam

Derivative of (sin(3x+2x))/(cos(3x-2x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
sin(3*x + 2*x)
--------------
cos(3*x - 2*x)
$$\frac{\sin{\left(2 x + 3 x \right)}}{\cos{\left(- 2 x + 3 x \right)}}$$
sin(3*x + 2*x)/cos(3*x - 2*x)
Detail solution
  1. Apply the quotient rule, which is:

    and .

    To find :

    1. Let .

    2. The derivative of sine is cosine:

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

      1. Differentiate term by term:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      The result of the chain rule is:

    To find :

    1. Let .

    2. The derivative of cosine is negative sine:

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

      1. Differentiate term by term:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      The result of the chain rule is:

    Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
5*cos(3*x + 2*x)   sin(3*x - 2*x)*sin(3*x + 2*x)
---------------- + -----------------------------
 cos(3*x - 2*x)              2                  
                          cos (3*x - 2*x)       
$$\frac{\sin{\left(- 2 x + 3 x \right)} \sin{\left(2 x + 3 x \right)}}{\cos^{2}{\left(- 2 x + 3 x \right)}} + \frac{5 \cos{\left(2 x + 3 x \right)}}{\cos{\left(- 2 x + 3 x \right)}}$$
The second derivative [src]
               /         2   \                              
               |    2*sin (x)|            10*cos(5*x)*sin(x)
-25*sin(5*x) + |1 + ---------|*sin(5*x) + ------------------
               |        2    |                  cos(x)      
               \     cos (x) /                              
------------------------------------------------------------
                           cos(x)                           
$$\frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(5 x \right)} + \frac{10 \sin{\left(x \right)} \cos{\left(5 x \right)}}{\cos{\left(x \right)}} - 25 \sin{\left(5 x \right)}}{\cos{\left(x \right)}}$$
The third derivative [src]
                                                                   /         2   \                
                                                                   |    6*sin (x)|                
                                                                   |5 + ---------|*sin(x)*sin(5*x)
                   /         2   \                                 |        2    |                
                   |    2*sin (x)|            75*sin(x)*sin(5*x)   \     cos (x) /                
-125*cos(5*x) + 15*|1 + ---------|*cos(5*x) - ------------------ + -------------------------------
                   |        2    |                  cos(x)                      cos(x)            
                   \     cos (x) /                                                                
--------------------------------------------------------------------------------------------------
                                              cos(x)                                              
$$\frac{15 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cos{\left(5 x \right)} + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)} \sin{\left(5 x \right)}}{\cos{\left(x \right)}} - \frac{75 \sin{\left(x \right)} \sin{\left(5 x \right)}}{\cos{\left(x \right)}} - 125 \cos{\left(5 x \right)}}{\cos{\left(x \right)}}$$