Mister Exam

Derivative of xsin^2x

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
     2   
x*sin (x)
$$x \sin^{2}{\left(x \right)}$$
d /     2   \
--\x*sin (x)/
dx           
$$\frac{d}{d x} x \sin^{2}{\left(x \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the power rule: goes to

    ; to find :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. The derivative of sine is cosine:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
   2                       
sin (x) + 2*x*cos(x)*sin(x)
$$2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}$$
The second derivative [src]
  /    /   2         2   \                  \
2*\- x*\sin (x) - cos (x)/ + 2*cos(x)*sin(x)/
$$2 \left(- x \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
The third derivative [src]
  /       2           2                       \
2*\- 3*sin (x) + 3*cos (x) - 4*x*cos(x)*sin(x)/
$$2 \left(- 4 x \sin{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right)$$
The graph
Derivative of xsin^2x