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x^4*cos(x)

Derivative of x^4*cos(x)

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

You have entered [src]
 4       
x *cos(x)
x4cos(x)x^{4} \cos{\left(x \right)}
x^4*cos(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)=cos(x)g{\left(x \right)} = \cos{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos(x)=sin(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    The result is: x4sin(x)+4x3cos(x)- x^{4} \sin{\left(x \right)} + 4 x^{3} \cos{\left(x \right)}

  2. Now simplify:

    x3(xsin(x)+4cos(x))x^{3} \left(- x \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)


The answer is:

x3(xsin(x)+4cos(x))x^{3} \left(- x \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right)

The graph
02468-8-6-4-2-1010-2000020000
The first derivative [src]
   4             3       
- x *sin(x) + 4*x *cos(x)
x4sin(x)+4x3cos(x)- x^{4} \sin{\left(x \right)} + 4 x^{3} \cos{\left(x \right)}
The second derivative [src]
 2 /             2                    \
x *\12*cos(x) - x *cos(x) - 8*x*sin(x)/
x2(x2cos(x)8xsin(x)+12cos(x))x^{2} \left(- x^{2} \cos{\left(x \right)} - 8 x \sin{\left(x \right)} + 12 \cos{\left(x \right)}\right)
The third derivative [src]
  /             3                            2       \
x*\24*cos(x) + x *sin(x) - 36*x*sin(x) - 12*x *cos(x)/
x(x3sin(x)12x2cos(x)36xsin(x)+24cos(x))x \left(x^{3} \sin{\left(x \right)} - 12 x^{2} \cos{\left(x \right)} - 36 x \sin{\left(x \right)} + 24 \cos{\left(x \right)}\right)
The graph
Derivative of x^4*cos(x)