Mister Exam

Derivative of y=x^3sinx*lnx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 3              
x *sin(x)*log(x)
x3log(x)sin(x)x^{3} \log{\left(x \right)} \sin{\left(x \right)}
d / 3              \
--\x *sin(x)*log(x)/
dx                  
ddxx3log(x)sin(x)\frac{d}{d x} x^{3} \log{\left(x \right)} \sin{\left(x \right)}
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)h(x)=f(x)g(x)ddxh(x)+f(x)h(x)ddxg(x)+g(x)h(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\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)}

    h(x)=log(x)h{\left(x \right)} = \log{\left(x \right)}; to find ddxh(x)\frac{d}{d x} h{\left(x \right)}:

    1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    The result is: x3log(x)cos(x)+3x2log(x)sin(x)+x2sin(x)x^{3} \log{\left(x \right)} \cos{\left(x \right)} + 3 x^{2} \log{\left(x \right)} \sin{\left(x \right)} + x^{2} \sin{\left(x \right)}

  2. Now simplify:

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


The answer is:

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

The graph
02468-8-6-4-2-1010-50005000
The first derivative [src]
 2           3                    2              
x *sin(x) + x *cos(x)*log(x) + 3*x *log(x)*sin(x)
x3log(x)cos(x)+3x2log(x)sin(x)+x2sin(x)x^{3} \log{\left(x \right)} \cos{\left(x \right)} + 3 x^{2} \log{\left(x \right)} \sin{\left(x \right)} + x^{2} \sin{\left(x \right)}
The second derivative [src]
  /                                           2                                  \
x*\5*sin(x) + 2*x*cos(x) + 6*log(x)*sin(x) - x *log(x)*sin(x) + 6*x*cos(x)*log(x)/
x(x2log(x)sin(x)+6xlog(x)cos(x)+2xcos(x)+6log(x)sin(x)+5sin(x))x \left(- x^{2} \log{\left(x \right)} \sin{\left(x \right)} + 6 x \log{\left(x \right)} \cos{\left(x \right)} + 2 x \cos{\left(x \right)} + 6 \log{\left(x \right)} \sin{\left(x \right)} + 5 \sin{\left(x \right)}\right)
3-th derivative [src]
               2                                           3                    2                                   
11*sin(x) - 3*x *sin(x) + 6*log(x)*sin(x) + 15*x*cos(x) - x *cos(x)*log(x) - 9*x *log(x)*sin(x) + 18*x*cos(x)*log(x)
x3log(x)cos(x)9x2log(x)sin(x)3x2sin(x)+18xlog(x)cos(x)+15xcos(x)+6log(x)sin(x)+11sin(x)- x^{3} \log{\left(x \right)} \cos{\left(x \right)} - 9 x^{2} \log{\left(x \right)} \sin{\left(x \right)} - 3 x^{2} \sin{\left(x \right)} + 18 x \log{\left(x \right)} \cos{\left(x \right)} + 15 x \cos{\left(x \right)} + 6 \log{\left(x \right)} \sin{\left(x \right)} + 11 \sin{\left(x \right)}
The third derivative [src]
               2                                           3                    2                                   
11*sin(x) - 3*x *sin(x) + 6*log(x)*sin(x) + 15*x*cos(x) - x *cos(x)*log(x) - 9*x *log(x)*sin(x) + 18*x*cos(x)*log(x)
x3log(x)cos(x)9x2log(x)sin(x)3x2sin(x)+18xlog(x)cos(x)+15xcos(x)+6log(x)sin(x)+11sin(x)- x^{3} \log{\left(x \right)} \cos{\left(x \right)} - 9 x^{2} \log{\left(x \right)} \sin{\left(x \right)} - 3 x^{2} \sin{\left(x \right)} + 18 x \log{\left(x \right)} \cos{\left(x \right)} + 15 x \cos{\left(x \right)} + 6 \log{\left(x \right)} \sin{\left(x \right)} + 11 \sin{\left(x \right)}
The graph
Derivative of y=x^3sinx*lnx