Derivative of sinsqrt(x)-sqrt(cosx)

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   /  ___\     ________
sin\\/ x / - \/ cos(x)

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

sin(sqrt(x)) - sqrt(cos(x))

Detail solution

Differentiate $\sin{\left(\sqrt{x} \right)} - \sqrt{\cos{\left(x \right)}}$ term by term:
1. Let $u = \sqrt{x}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. 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:
  
  $\frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$
4. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(x \right)}$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result of the chain rule is:
    
    $- \frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}}$
  So, the result is: $\frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}}$
The result is: $\frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} + \frac{\cos{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$

The answer is:

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

The graph

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The first derivative [src]

   /  ___\               
cos\\/ x /      sin(x)   
---------- + ------------
     ___         ________
 2*\/ x      2*\/ cos(x)

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

Simplify

The second derivative [src]

                   2          /  ___\      /  ___\
    ________    sin (x)    sin\\/ x /   cos\\/ x /
2*\/ cos(x)  + --------- - ---------- - ----------
                  3/2          x            3/2   
               cos   (x)                   x      
--------------------------------------------------
                        4

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sinsqrt(x)-sqrt(cosx)

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