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sin(x)^(1/2)
The derivative of the function/ sin(x)^(1/2)

Derivative of sin(x)^(1/2)

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

The graph:

from to

Piecewise:

The solution

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

  1. Let u=sin⁡(x)u = \sin{\left(x \right)}.

  2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

  3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-10105-10
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   cos(x)   
------------
    ________
2*\/ sin(x)
cos⁡(x)2sin⁡(x)\frac{\cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}
Simplify
The second derivative [src] 
 /                   2    \ 
 |    ________    cos (x) | 
-|2*\/ sin(x)  + ---------| 
 |                  3/2   | 
 \               sin   (x)/ 
----------------------------
             4
−2sin⁡(x)+cos⁡2(x)sin⁡32(x)4- \frac{2 \sqrt{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}}{4}
Simplify
The third derivative [src] 
/         2   \       
|    3*cos (x)|       
|2 + ---------|*cos(x)
|        2    |       
\     sin (x) /       
----------------------
         ________     
     8*\/ sin(x)
(2+3cos⁡2(x)sin⁡2(x))cos⁡(x)8sin⁡(x)\frac{\left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{8 \sqrt{\sin{\left(x \right)}}}
Simplify
The graph
Derivative of sin(x)^(1/2)
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