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y(x)=sinx√cos2x
The derivative of the function/ y(x)=sinx√cos2x

Derivative of y(x)=sinx√cos2x

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

You have entered [src] 
         __________
sin(x)*\/ cos(2*x)
sin⁡(x)cos⁡(2x)\sin{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} 
d /         __________\
--\sin(x)*\/ cos(2*x) /
dx
ddxsin⁡(x)cos⁡(2x)\frac{d}{d x} \sin{\left(x \right)} \sqrt{\cos{\left(2 x \right)}}
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    g(x)=cos⁡(2x)g{\left(x \right)} = \sqrt{\cos{\left(2 x \right)}}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=cos⁡(2x)u = \cos{\left(2 x \right)}.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddxcos⁡(2x)\frac{d}{d x} \cos{\left(2 x \right)}:

      1. Let u=2xu = 2 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 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:

        −2sin⁡(2x)- 2 \sin{\left(2 x \right)}

      The result of the chain rule is:

      −sin⁡(2x)cos⁡(2x)- \frac{\sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}

    The result is: −sin⁡(x)sin⁡(2x)cos⁡(2x)+cos⁡(x)cos⁡(2x)- \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}} + \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}}

  2. Now simplify:

    cos⁡(3x)cos⁡(2x)\frac{\cos{\left(3 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}

The answer is:

cos⁡(3x)cos⁡(2x)\frac{\cos{\left(3 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  __________          sin(x)*sin(2*x)
\/ cos(2*x) *cos(x) - ---------------
                          __________ 
                        \/ cos(2*x)
cos⁡(x)cos⁡(2x)−sin⁡(x)sin⁡(2x)cos⁡(2x)\cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} - \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}
Simplify
The second derivative [src] 
 /                      /                     2      \                           \
 |  __________          |    __________    sin (2*x) |          2*cos(x)*sin(2*x)|
-|\/ cos(2*x) *sin(x) + |2*\/ cos(2*x)  + -----------|*sin(x) + -----------------|
 |                      |                    3/2     |               __________  |
 \                      \                 cos   (2*x)/             \/ cos(2*x)   /
−((2cos⁡(2x)+sin⁡2(2x)cos⁡32(2x))sin⁡(x)+sin⁡(x)cos⁡(2x)+2sin⁡(2x)cos⁡(x)cos⁡(2x))- (\left(2 \sqrt{\cos{\left(2 x \right)}} + \frac{\sin^{2}{\left(2 x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}}\right) \sin{\left(x \right)} + \sin{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + \frac{2 \sin{\left(2 x \right)} \cos{\left(x \right)}}{\sqrt{\cos{\left(2 x \right)}}})
Simplify
The third derivative [src] 
                                                                                      /         2     \                
                                                                                      |    3*sin (2*x)|                
                                                                                      |2 + -----------|*sin(x)*sin(2*x)
                          /                     2      \                              |        2      |                
    __________            |    __________    sin (2*x) |          3*sin(x)*sin(2*x)   \     cos (2*x) /                
- \/ cos(2*x) *cos(x) - 3*|2*\/ cos(2*x)  + -----------|*cos(x) + ----------------- - ---------------------------------
                          |                    3/2     |               __________                  __________          
                          \                 cos   (2*x)/             \/ cos(2*x)                 \/ cos(2*x)
−(3sin⁡2(2x)cos⁡2(2x)+2)sin⁡(x)sin⁡(2x)cos⁡(2x)−3(2cos⁡(2x)+sin⁡2(2x)cos⁡32(2x))cos⁡(x)−cos⁡(x)cos⁡(2x)+3sin⁡(x)sin⁡(2x)cos⁡(2x)- \frac{\left(\frac{3 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 2\right) \sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}} - 3 \left(2 \sqrt{\cos{\left(2 x \right)}} + \frac{\sin^{2}{\left(2 x \right)}}{\cos^{\frac{3}{2}}{\left(2 x \right)}}\right) \cos{\left(x \right)} - \cos{\left(x \right)} \sqrt{\cos{\left(2 x \right)}} + \frac{3 \sin{\left(x \right)} \sin{\left(2 x \right)}}{\sqrt{\cos{\left(2 x \right)}}}
Simplify
The graph
Derivative of y(x)=sinx√cos2x
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