Mister Exam

Derivative of y=x*sin(x)*ln(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
x*sin(x)*log(x)
$$x \sin{\left(x \right)} \log{\left(x \right)}$$
(x*sin(x))*log(x)
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the product rule:

      ; to find :

      1. Apply the power rule: goes to

      ; to find :

      1. The derivative of sine is cosine:

      The result is:

    ; to find :

    1. The derivative of is .

    The result is:


The answer is:

The graph
The first derivative [src]
(x*cos(x) + sin(x))*log(x) + sin(x)
$$\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \log{\left(x \right)} + \sin{\left(x \right)}$$
The second derivative [src]
  sin(x)                                   2*(x*cos(x) + sin(x))
- ------ - (-2*cos(x) + x*sin(x))*log(x) + ---------------------
    x                                                x          
$$- \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \log{\left(x \right)} + \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x} - \frac{\sin{\left(x \right)}}{x}$$
The third derivative [src]
                                3*(-2*cos(x) + x*sin(x))   3*(x*cos(x) + sin(x))   2*sin(x)
-(3*sin(x) + x*cos(x))*log(x) - ------------------------ - --------------------- + --------
                                           x                          2                2   
                                                                     x                x    
$$- \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \log{\left(x \right)} - \frac{3 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right)}{x} - \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{2}}$$
The graph
Derivative of y=x*sin(x)*ln(x)