Mister Exam

Derivative of (0,5-x)cosx+sinx

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
(1/2 - x)*cos(x) + sin(x)
$$\left(\frac{1}{2} - x\right) \cos{\left(x \right)} + \sin{\left(x \right)}$$
(1/2 - x)*cos(x) + sin(x)
Detail solution
  1. Differentiate term by term:

    1. Apply the quotient rule, which is:

      and .

      To find :

      1. Apply the product rule:

        ; to find :

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

          2. 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 is:

        ; to find :

        1. The derivative of cosine is negative sine:

        The result is:

      To find :

      1. The derivative of the constant is zero.

      Now plug in to the quotient rule:

    2. The derivative of sine is cosine:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
-(1/2 - x)*sin(x)
$$- \left(\frac{1}{2} - x\right) \sin{\left(x \right)}$$
The second derivative [src]
(-1 + 2*x)*cos(x)         
----------------- + sin(x)
        2                 
$$\frac{\left(2 x - 1\right) \cos{\left(x \right)}}{2} + \sin{\left(x \right)}$$
The third derivative [src]
           (-1 + 2*x)*sin(x)
2*cos(x) - -----------------
                   2        
$$- \frac{\left(2 x - 1\right) \sin{\left(x \right)}}{2} + 2 \cos{\left(x \right)}$$
The graph
Derivative of (0,5-x)cosx+sinx