Mister Exam

Derivative of xcos(pix)

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

The graph:

from to

Piecewise:

The solution

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

    ; to find :

    1. Apply the power rule: goes to

    ; to find :

    1. Let .

    2. The derivative of cosine is negative sine:

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

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result of the chain rule is:

    The result is:


The answer is:

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