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sin(2*x)/cos(5*x)

Derivative of sin(2*x)/cos(5*x)

Function f() - derivative -N order at the point
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The solution

You have entered [src]
sin(2*x)
--------
cos(5*x)
sin(2x)cos(5x)\frac{\sin{\left(2 x \right)}}{\cos{\left(5 x \right)}}
sin(2*x)/cos(5*x)
Detail solution
  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin(2x)f{\left(x \right)} = \sin{\left(2 x \right)} and g(x)=cos(5x)g{\left(x \right)} = \cos{\left(5 x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=2xu = 2 x.

    2. The derivative of sine is cosine:

      ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

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

        1. Apply the power rule: xx goes to 11

        So, the result is: 22

      The result of the chain rule is:

      2cos(2x)2 \cos{\left(2 x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=5xu = 5 x.

    2. The derivative of cosine is negative sine:

      dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddx5x\frac{d}{d x} 5 x:

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

        1. Apply the power rule: xx goes to 11

        So, the result is: 55

      The result of the chain rule is:

      5sin(5x)- 5 \sin{\left(5 x \right)}

    Now plug in to the quotient rule:

    5sin(2x)sin(5x)+2cos(2x)cos(5x)cos2(5x)\frac{5 \sin{\left(2 x \right)} \sin{\left(5 x \right)} + 2 \cos{\left(2 x \right)} \cos{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}

  2. Now simplify:

    7cos(3x)3cos(7x)2cos2(5x)\frac{7 \cos{\left(3 x \right)} - 3 \cos{\left(7 x \right)}}{2 \cos^{2}{\left(5 x \right)}}


The answer is:

7cos(3x)3cos(7x)2cos2(5x)\frac{7 \cos{\left(3 x \right)} - 3 \cos{\left(7 x \right)}}{2 \cos^{2}{\left(5 x \right)}}

The graph
02468-8-6-4-2-1010-25002500
The first derivative [src]
2*cos(2*x)   5*sin(2*x)*sin(5*x)
---------- + -------------------
 cos(5*x)            2          
                  cos (5*x)     
5sin(2x)sin(5x)cos2(5x)+2cos(2x)cos(5x)\frac{5 \sin{\left(2 x \right)} \sin{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + \frac{2 \cos{\left(2 x \right)}}{\cos{\left(5 x \right)}}
The second derivative [src]
                 /         2     \                                
                 |    2*sin (5*x)|            20*cos(2*x)*sin(5*x)
-4*sin(2*x) + 25*|1 + -----------|*sin(2*x) + --------------------
                 |        2      |                  cos(5*x)      
                 \     cos (5*x) /                                
------------------------------------------------------------------
                             cos(5*x)                             
25(2sin2(5x)cos2(5x)+1)sin(2x)4sin(2x)+20sin(5x)cos(2x)cos(5x)cos(5x)\frac{25 \left(\frac{2 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \sin{\left(2 x \right)} - 4 \sin{\left(2 x \right)} + \frac{20 \sin{\left(5 x \right)} \cos{\left(2 x \right)}}{\cos{\left(5 x \right)}}}{\cos{\left(5 x \right)}}
The third derivative [src]
                                                                          /         2     \                  
                                                                          |    6*sin (5*x)|                  
                                                                      125*|5 + -----------|*sin(2*x)*sin(5*x)
                  /         2     \                                       |        2      |                  
                  |    2*sin (5*x)|            60*sin(2*x)*sin(5*x)       \     cos (5*x) /                  
-8*cos(2*x) + 150*|1 + -----------|*cos(2*x) - -------------------- + ---------------------------------------
                  |        2      |                  cos(5*x)                         cos(5*x)               
                  \     cos (5*x) /                                                                          
-------------------------------------------------------------------------------------------------------------
                                                   cos(5*x)                                                  
150(2sin2(5x)cos2(5x)+1)cos(2x)+125(6sin2(5x)cos2(5x)+5)sin(2x)sin(5x)cos(5x)60sin(2x)sin(5x)cos(5x)8cos(2x)cos(5x)\frac{150 \left(\frac{2 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \cos{\left(2 x \right)} + \frac{125 \left(\frac{6 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 5\right) \sin{\left(2 x \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - \frac{60 \sin{\left(2 x \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}} - 8 \cos{\left(2 x \right)}}{\cos{\left(5 x \right)}}
The graph
Derivative of sin(2*x)/cos(5*x)