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y=x^4*sinsinx
The derivative of the function/ y=x^4*sinsinx

Derivative of y=x^4*sinsinx

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

The graph:

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The solution

You have entered [src] 
 4            
x *sin(sin(x))
x4sin⁡(sin⁡(x))x^{4} \sin{\left(\sin{\left(x \right)} \right)} 
x^4*sin(sin(x))
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)=x4f{\left(x \right)} = x^{4}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

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

    1. Let u=sin⁡(x)u = \sin{\left(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 ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

      The result of the chain rule is:

      cos⁡(x)cos⁡(sin⁡(x))\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}

    The result is: x4cos⁡(x)cos⁡(sin⁡(x))+4x3sin⁡(sin⁡(x))x^{4} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 4 x^{3} \sin{\left(\sin{\left(x \right)} \right)}

  2. Now simplify:

    x3(xcos⁡(x)cos⁡(sin⁡(x))+4sin⁡(sin⁡(x)))x^{3} \left(x \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)}\right)

The answer is:

x3(xcos⁡(x)cos⁡(sin⁡(x))+4sin⁡(sin⁡(x)))x^{3} \left(x \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)}\right)

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   3                4                   
4*x *sin(sin(x)) + x *cos(x)*cos(sin(x))
x4cos⁡(x)cos⁡(sin⁡(x))+4x3sin⁡(sin⁡(x))x^{4} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 4 x^{3} \sin{\left(\sin{\left(x \right)} \right)}
Simplify
The second derivative [src] 
 2 /                  2 /   2                                    \                         \
x *\12*sin(sin(x)) - x *\cos (x)*sin(sin(x)) + cos(sin(x))*sin(x)/ + 8*x*cos(x)*cos(sin(x))/
x2(−x2(sin⁡(x)cos⁡(sin⁡(x))+sin⁡(sin⁡(x))cos⁡2(x))+8xcos⁡(x)cos⁡(sin⁡(x))+12sin⁡(sin⁡(x)))x^{2} \left(- x^{2} \left(\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right) + 8 x \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 12 \sin{\left(\sin{\left(x \right)} \right)}\right)
Simplify
The third derivative [src] 
  /                     2 /   2                                    \    3 /   2                                                    \                                 \
x*\24*sin(sin(x)) - 12*x *\cos (x)*sin(sin(x)) + cos(sin(x))*sin(x)/ - x *\cos (x)*cos(sin(x)) - 3*sin(x)*sin(sin(x)) + cos(sin(x))/*cos(x) + 36*x*cos(x)*cos(sin(x))/
x(−x3(−3sin⁡(x)sin⁡(sin⁡(x))+cos⁡2(x)cos⁡(sin⁡(x))+cos⁡(sin⁡(x)))cos⁡(x)−12x2(sin⁡(x)cos⁡(sin⁡(x))+sin⁡(sin⁡(x))cos⁡2(x))+36xcos⁡(x)cos⁡(sin⁡(x))+24sin⁡(sin⁡(x)))x \left(- x^{3} \left(- 3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} + \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} - 12 x^{2} \left(\sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)}\right) + 36 x \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + 24 \sin{\left(\sin{\left(x \right)} \right)}\right)
Simplify
The graph
Derivative of y=x^4*sinsinx
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