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y=sin(0.5*x)
The derivative of the function/ y=sin(0.5*x)

Derivative of y=sin(0.5*x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /x\
sin|-|
   \2/
sin⁡(x2)\sin{\left(\frac{x}{2} \right)} 
d /   /x\\
--|sin|-||
dx\   \2//
ddxsin⁡(x2)\frac{d}{d x} \sin{\left(\frac{x}{2} \right)}
Transform
Detail solution

  1. Let u=x2u = \frac{x}{2}.

  2. The derivative of sine is cosine:

    ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddxx2\frac{d}{d x} \frac{x}{2}:

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

      1. Apply the power rule: xx goes to 11

      So, the result is: 12\frac{1}{2}

    The result of the chain rule is:

    cos⁡(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}

The answer is:

cos⁡(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   /x\
cos|-|
   \2/
------
  2
cos⁡(x2)2\frac{\cos{\left(\frac{x}{2} \right)}}{2}
Simplify
The second derivative [src] 
    /x\ 
-sin|-| 
    \2/ 
--------
   4
−sin⁡(x2)4- \frac{\sin{\left(\frac{x}{2} \right)}}{4}
Simplify
The third derivative [src] 
    /x\ 
-cos|-| 
    \2/ 
--------
   8
−cos⁡(x2)8- \frac{\cos{\left(\frac{x}{2} \right)}}{8}
Simplify
The graph
Derivative of y=sin(0.5*x)
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