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sin(sqrt(x))
The derivative of the function/ sin(sqrt(x))

Derivative of sin(sqrt(x))

Function f() - derivative -N order at the point
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The graph:

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The solution

You have entered [src] 
   /  ___\
sin\\/ x /
sin⁡(x)\sin{\left(\sqrt{x} \right)} 
sin(sqrt(x))
Detail solution

  1. Let u=xu = \sqrt{x}.

  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 ddxx\frac{d}{d x} \sqrt{x}:

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

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   /  ___\
cos\\/ x /
----------
     ___  
 2*\/ x
cos⁡(x)2x\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
 /   /  ___\      /  ___\\ 
 |sin\\/ x /   cos\\/ x /| 
-|---------- + ----------| 
 |    x            3/2   | 
 \                x      / 
---------------------------
             4
−sin⁡(x)x+cos⁡(x)x324- \frac{\frac{\sin{\left(\sqrt{x} \right)}}{x} + \frac{\cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}}{4}
Simplify
The third derivative [src] 
     /  ___\        /  ___\        /  ___\
  cos\\/ x /   3*sin\\/ x /   3*cos\\/ x /
- ---------- + ------------ + ------------
      3/2            2             5/2    
     x              x             x       
------------------------------------------
                    8
3sin⁡(x)x2−cos⁡(x)x32+3cos⁡(x)x528\frac{\frac{3 \sin{\left(\sqrt{x} \right)}}{x^{2}} - \frac{\cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} + \frac{3 \cos{\left(\sqrt{x} \right)}}{x^{\frac{5}{2}}}}{8}
Simplify
The graph
Derivative of sin(sqrt(x))
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