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The derivative of the function/ y=sin^n*cosnx

Derivative of y=sin^n*cosnx

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

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The solution

You have entered [src] 
   n          
sin (cos(n*x))
sin⁡n(cos⁡(nx))\sin^{n}{\left(\cos{\left(n x \right)} \right)} 
sin(cos(n*x))^n
Detail solution

  1. Let u=sin⁡(cos⁡(nx))u = \sin{\left(\cos{\left(n x \right)} \right)}.

  2. Apply the power rule: unu^{n} goes to nunu\frac{n u^{n}}{u}

  3. Then, apply the chain rule. Multiply by ∂∂xsin⁡(cos⁡(nx))\frac{\partial}{\partial x} \sin{\left(\cos{\left(n x \right)} \right)}:

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

    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 ∂∂xcos⁡(nx)\frac{\partial}{\partial x} \cos{\left(n x \right)}:

      1. Let u=nxu = n 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 ∂∂xnx\frac{\partial}{\partial x} n 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: nn

        The result of the chain rule is:

        −nsin⁡(nx)- n \sin{\left(n x \right)}

      The result of the chain rule is:

      −nsin⁡(nx)cos⁡(cos⁡(nx))- n \sin{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}

    The result of the chain rule is:

    −n2sin⁡(nx)sin⁡n(cos⁡(nx))cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))- \frac{n^{2} \sin{\left(n x \right)} \sin^{n}{\left(\cos{\left(n x \right)} \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}

  4. Now simplify:

    −n2sin⁡(nx)sin⁡n−1(cos⁡(nx))cos⁡(cos⁡(nx))- n^{2} \sin{\left(n x \right)} \sin^{n - 1}{\left(\cos{\left(n x \right)} \right)} \cos{\left(\cos{\left(n x \right)} \right)}

The answer is:

−n2sin⁡(nx)sin⁡n−1(cos⁡(nx))cos⁡(cos⁡(nx))- n^{2} \sin{\left(n x \right)} \sin^{n - 1}{\left(\cos{\left(n x \right)} \right)} \cos{\left(\cos{\left(n x \right)} \right)}

The first derivative [src] 
  2    n                                  
-n *sin (cos(n*x))*cos(cos(n*x))*sin(n*x) 
------------------------------------------
              sin(cos(n*x))
−n2sin⁡(nx)sin⁡n(cos⁡(nx))cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))- \frac{n^{2} \sin{\left(n x \right)} \sin^{n}{\left(\cos{\left(n x \right)} \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}
Simplify
The second derivative [src] 
                  /                 2              2                                      2              2     \
 3    n           |     2        cos (cos(n*x))*sin (n*x)   cos(n*x)*cos(cos(n*x))   n*cos (cos(n*x))*sin (n*x)|
n *sin (cos(n*x))*|- sin (n*x) - ------------------------ - ---------------------- + --------------------------|
                  |                      2                      sin(cos(n*x))                 2                |
                  \                   sin (cos(n*x))                                       sin (cos(n*x))      /
n3(nsin⁡2(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))−sin⁡2(nx)−sin⁡2(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))−cos⁡(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx)))sin⁡n(cos⁡(nx))n^{3} \left(\frac{n \sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \sin^{2}{\left(n x \right)} - \frac{\sin^{2}{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \frac{\cos{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}}\right) \sin^{n}{\left(\cos{\left(n x \right)} \right)}
Simplify
The third derivative [src] 
                  /                                   2                           3              2             2                       2    3              2               2                             3              2               2                   \         
 4    n           |              cos(cos(n*x))   3*cos (cos(n*x))*cos(n*x)   2*cos (cos(n*x))*sin (n*x)   2*sin (n*x)*cos(cos(n*x))   n *cos (cos(n*x))*sin (n*x)   3*n*cos (cos(n*x))*cos(n*x)   3*n*cos (cos(n*x))*sin (n*x)   3*n*sin (n*x)*cos(cos(n*x))|         
n *sin (cos(n*x))*|-3*cos(n*x) + ------------- - ------------------------- - -------------------------- - ------------------------- - --------------------------- + --------------------------- + ---------------------------- + ---------------------------|*sin(n*x)
                  |              sin(cos(n*x))            2                           3                         sin(cos(n*x))                   3                             2                             3                           sin(cos(n*x))       |         
                  \                                    sin (cos(n*x))              sin (cos(n*x))                                            sin (cos(n*x))                sin (cos(n*x))                sin (cos(n*x))                                     /
n4(−n2sin⁡2(nx)cos⁡3(cos⁡(nx))sin⁡3(cos⁡(nx))+3nsin⁡2(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))+3nsin⁡2(nx)cos⁡3(cos⁡(nx))sin⁡3(cos⁡(nx))+3ncos⁡(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx))−2sin⁡2(nx)cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−2sin⁡2(nx)cos⁡3(cos⁡(nx))sin⁡3(cos⁡(nx))−3cos⁡(nx)+cos⁡(cos⁡(nx))sin⁡(cos⁡(nx))−3cos⁡(nx)cos⁡2(cos⁡(nx))sin⁡2(cos⁡(nx)))sin⁡(nx)sin⁡n(cos⁡(nx))n^{4} \left(- \frac{n^{2} \sin^{2}{\left(n x \right)} \cos^{3}{\left(\cos{\left(n x \right)} \right)}}{\sin^{3}{\left(\cos{\left(n x \right)} \right)}} + \frac{3 n \sin^{2}{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} + \frac{3 n \sin^{2}{\left(n x \right)} \cos^{3}{\left(\cos{\left(n x \right)} \right)}}{\sin^{3}{\left(\cos{\left(n x \right)} \right)}} + \frac{3 n \cos{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}} - \frac{2 \sin^{2}{\left(n x \right)} \cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \frac{2 \sin^{2}{\left(n x \right)} \cos^{3}{\left(\cos{\left(n x \right)} \right)}}{\sin^{3}{\left(\cos{\left(n x \right)} \right)}} - 3 \cos{\left(n x \right)} + \frac{\cos{\left(\cos{\left(n x \right)} \right)}}{\sin{\left(\cos{\left(n x \right)} \right)}} - \frac{3 \cos{\left(n x \right)} \cos^{2}{\left(\cos{\left(n x \right)} \right)}}{\sin^{2}{\left(\cos{\left(n x \right)} \right)}}\right) \sin{\left(n x \right)} \sin^{n}{\left(\cos{\left(n x \right)} \right)}
Simplify
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