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The derivative of the function/ 50*sin(y*cos(y))^(3)

Derivative of 50*sin(y*cos(y))^(3)

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

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      3          
50*sin (y*cos(y))
50sin⁡3(ycos⁡(y))50 \sin^{3}{\left(y \cos{\left(y \right)} \right)} 
50*sin(y*cos(y))^3
Detail solution

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

    1. Let u=sin⁡(ycos⁡(y))u = \sin{\left(y \cos{\left(y \right)} \right)}.

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    3. Then, apply the chain rule. Multiply by ddysin⁡(ycos⁡(y))\frac{d}{d y} \sin{\left(y \cos{\left(y \right)} \right)}:

      1. Let u=ycos⁡(y)u = y \cos{\left(y \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 ddyycos⁡(y)\frac{d}{d y} y \cos{\left(y \right)}:

        1. Apply the product rule:

          ddyf(y)g(y)=f(y)ddyg(y)+g(y)ddyf(y)\frac{d}{d y} f{\left(y \right)} g{\left(y \right)} = f{\left(y \right)} \frac{d}{d y} g{\left(y \right)} + g{\left(y \right)} \frac{d}{d y} f{\left(y \right)}

          f(y)=yf{\left(y \right)} = y; to find ddyf(y)\frac{d}{d y} f{\left(y \right)}:

          1. Apply the power rule: yy goes to 11

          g(y)=cos⁡(y)g{\left(y \right)} = \cos{\left(y \right)}; to find ddyg(y)\frac{d}{d y} g{\left(y \right)}:

          1. The derivative of cosine is negative sine:

            ddycos⁡(y)=−sin⁡(y)\frac{d}{d y} \cos{\left(y \right)} = - \sin{\left(y \right)}

          The result is: −ysin⁡(y)+cos⁡(y)- y \sin{\left(y \right)} + \cos{\left(y \right)}

        The result of the chain rule is:

        (−ysin⁡(y)+cos⁡(y))cos⁡(ycos⁡(y))\left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \cos{\left(y \cos{\left(y \right)} \right)}

      The result of the chain rule is:

      3(−ysin⁡(y)+cos⁡(y))sin⁡2(ycos⁡(y))cos⁡(ycos⁡(y))3 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}

    So, the result is: 150(−ysin⁡(y)+cos⁡(y))sin⁡2(ycos⁡(y))cos⁡(ycos⁡(y))150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}

The answer is:

150(−ysin⁡(y)+cos⁡(y))sin⁡2(ycos⁡(y))cos⁡(ycos⁡(y))150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}

The graph
02468-8-6-4-2-1010-10001000
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The first derivative [src] 
       2                                             
150*sin (y*cos(y))*(-y*sin(y) + cos(y))*cos(y*cos(y))
150(−ysin⁡(y)+cos⁡(y))sin⁡2(ycos⁡(y))cos⁡(ycos⁡(y))150 \left(- y \sin{\left(y \right)} + \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}
Simplify
The second derivative [src] 
     /                    2    2                                   2    2                                                              \              
-150*\(-cos(y) + y*sin(y)) *sin (y*cos(y)) - 2*(-cos(y) + y*sin(y)) *cos (y*cos(y)) + (2*sin(y) + y*cos(y))*cos(y*cos(y))*sin(y*cos(y))/*sin(y*cos(y))
−150((ysin⁡(y)−cos⁡(y))2sin⁡2(ycos⁡(y))−2(ysin⁡(y)−cos⁡(y))2cos⁡2(ycos⁡(y))+(ycos⁡(y)+2sin⁡(y))sin⁡(ycos⁡(y))cos⁡(ycos⁡(y)))sin⁡(ycos⁡(y))- 150 \left(\left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{2} \sin^{2}{\left(y \cos{\left(y \right)} \right)} - 2 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{2} \cos^{2}{\left(y \cos{\left(y \right)} \right)} + \left(y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)}\right) \sin{\left(y \cos{\left(y \right)} \right)}
Simplify
The third derivative [src] 
    /                        3    3                2                                                       3                                                                              3    2                                2                                                                   \
150*\- 2*(-cos(y) + y*sin(y)) *cos (y*cos(y)) + sin (y*cos(y))*(-3*cos(y) + y*sin(y))*cos(y*cos(y)) - 3*sin (y*cos(y))*(-cos(y) + y*sin(y))*(2*sin(y) + y*cos(y)) + 7*(-cos(y) + y*sin(y)) *sin (y*cos(y))*cos(y*cos(y)) + 6*cos (y*cos(y))*(-cos(y) + y*sin(y))*(2*sin(y) + y*cos(y))*sin(y*cos(y))/
150((ysin⁡(y)−3cos⁡(y))sin⁡2(ycos⁡(y))cos⁡(ycos⁡(y))+7(ysin⁡(y)−cos⁡(y))3sin⁡2(ycos⁡(y))cos⁡(ycos⁡(y))−2(ysin⁡(y)−cos⁡(y))3cos⁡3(ycos⁡(y))−3(ysin⁡(y)−cos⁡(y))(ycos⁡(y)+2sin⁡(y))sin⁡3(ycos⁡(y))+6(ysin⁡(y)−cos⁡(y))(ycos⁡(y)+2sin⁡(y))sin⁡(ycos⁡(y))cos⁡2(ycos⁡(y)))150 \left(\left(y \sin{\left(y \right)} - 3 \cos{\left(y \right)}\right) \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)} + 7 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{3} \sin^{2}{\left(y \cos{\left(y \right)} \right)} \cos{\left(y \cos{\left(y \right)} \right)} - 2 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right)^{3} \cos^{3}{\left(y \cos{\left(y \right)} \right)} - 3 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right) \left(y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin^{3}{\left(y \cos{\left(y \right)} \right)} + 6 \left(y \sin{\left(y \right)} - \cos{\left(y \right)}\right) \left(y \cos{\left(y \right)} + 2 \sin{\left(y \right)}\right) \sin{\left(y \cos{\left(y \right)} \right)} \cos^{2}{\left(y \cos{\left(y \right)} \right)}\right)
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