Derivative of sin(sqrt(x))

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   /  ___\
sin\\/ x /

$$\sin{\left(\sqrt{x} \right)}$$

sin(sqrt(x))

Detail solution

Let $u = \sqrt{x}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   /  ___\
cos\\/ x /
----------
     ___  
 2*\/ x

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

Simplify

The second derivative [src]

 /   /  ___\      /  ___\\ 
 |sin\\/ x /   cos\\/ x /| 
-|---------- + ----------| 
 |    x            3/2   | 
 \                x      / 
---------------------------
             4

$$- \frac{\frac{\sin{\left(\sqrt{x} \right)}}{x} + \frac{\cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}}}{4}$$

Simplify

The third derivative [src]

     /  ___\        /  ___\        /  ___\
  cos\\/ x /   3*sin\\/ x /   3*cos\\/ x /
- ---------- + ------------ + ------------
      3/2            2             5/2    
     x              x             x       
------------------------------------------
                    8

$$\frac{\frac{3 \sin{\left(\sqrt{x} \right)}}{x^{2}} - \frac{\cos{\left(\sqrt{x} \right)}}{x^{\frac{3}{2}}} + \frac{3 \cos{\left(\sqrt{x} \right)}}{x^{\frac{5}{2}}}}{8}$$

Simplify

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Derivative of sin(sqrt(x))

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The solution