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y=(x^3)(sinx)

Derivative of y=(x^3)(sinx)

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

You have entered [src]
 3       
x *sin(x)
x3sin(x)x^{3} \sin{\left(x \right)}
d / 3       \
--\x *sin(x)/
dx           
ddxx3sin(x)\frac{d}{d x} x^{3} \sin{\left(x \right)}
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)=x3f{\left(x \right)} = x^{3}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

    g(x)=sin(x)g{\left(x \right)} = \sin{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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 is: x3cos(x)+3x2sin(x)x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}

  2. Now simplify:

    x2(xcos(x)+3sin(x))x^{2} \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right)


The answer is:

x2(xcos(x)+3sin(x))x^{2} \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right)

The graph
02468-8-6-4-2-1010-20002000
The first derivative [src]
 3             2       
x *cos(x) + 3*x *sin(x)
x3cos(x)+3x2sin(x)x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}
The second derivative [src]
  /            2                    \
x*\6*sin(x) - x *sin(x) + 6*x*cos(x)/
x(x2sin(x)+6xcos(x)+6sin(x))x \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right)
The third derivative [src]
            3             2                     
6*sin(x) - x *cos(x) - 9*x *sin(x) + 18*x*cos(x)
x3cos(x)9x2sin(x)+18xcos(x)+6sin(x)- x^{3} \cos{\left(x \right)} - 9 x^{2} \sin{\left(x \right)} + 18 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}
The graph
Derivative of y=(x^3)(sinx)