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

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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

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(2 x + 3 x \right)}$ and $g{\left(x \right)} = \cos{\left(- 2 x + 3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x + 3 x$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 3 x\right)$ :
  1. Differentiate $2 x + 3 x$ 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: $x$ goes to $1$
      So, the result is: $3$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result is: $5$
  The result of the chain rule is:
  
  $5 \cos{\left(2 x + 3 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = - 2 x + 3 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 2 x + 3 x\right)$ :
  1. Differentiate $- 2 x + 3 x$ 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: $x$ goes to $1$
      So, the result is: $3$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $-2$
    The result is: $1$
  The result of the chain rule is:
  
  $- \sin{\left(- 2 x + 3 x \right)}$
Now plug in to the quotient rule:

$\frac{\sin{\left(- 2 x + 3 x \right)} \sin{\left(2 x + 3 x \right)} + 5 \cos{\left(- 2 x + 3 x \right)} \cos{\left(2 x + 3 x \right)}}{\cos^{2}{\left(- 2 x + 3 x \right)}}$
Now simplify:

$\frac{3 \cos{\left(4 x \right)} + 2 \cos{\left(6 x \right)}}{\cos^{2}{\left(x \right)}}$

The answer is:

\frac{3 \cos{\left(4 x \right)} + 2 \cos{\left(6 x \right)}}{\cos^{2}{\left(x \right)}}

The graph

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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)}}

Simplify

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)}}

Simplify

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)}}

Simplify

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

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