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Identical expressions
sqrt(one +x*sinx)-sqrt(cosx)
square root of (1 plus x multiply by sinus of x) minus square root of ( co sinus of e of x)
square root of (one plus x multiply by sinus of x) minus square root of ( co sinus of e of x)
√(1+x*sinx)-√(cosx)
sqrt(1+xsinx)-sqrt(cosx)
sqrt1+xsinx-sqrtcosx
Similar expressions
sqrt(1+x*sinx)+sqrt(cosx)
sqrt(1-x*sinx)-sqrt(cosx)

The derivative of the function/ sqrt(1+x*sinx)-sqrt(cosx)

Derivative of sqrt(1+x*sinx)-sqrt(cosx)

The solution

You have entered [src]

  ______________     ________
\/ 1 + x*sin(x)  - \/ cos(x)

$$\sqrt{x \sin{\left(x \right)} + 1} - \sqrt{\cos{\left(x \right)}}$$

d /  ______________     ________\
--\\/ 1 + x*sin(x)  - \/ cos(x) /
dx

$$\frac{d}{d x} \left(\sqrt{x \sin{\left(x \right)} + 1} - \sqrt{\cos{\left(x \right)}}\right)$$

Transform

Detail solution

Differentiate $\sqrt{x \sin{\left(x \right)} + 1} - \sqrt{\cos{\left(x \right)}}$ term by term:
1. Let $u = x \sin{\left(x \right)} + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x \sin{\left(x \right)} + 1\right)$ :
  1. Differentiate $x \sin{\left(x \right)} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      The result is: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
    The result is: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + 1}}$
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: $\frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + 1}}$
Now simplify:

$\frac{x \cos{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + 1}} + \frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} + \frac{\sin{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + 1}}$

The answer is:

$\frac{x \cos{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + 1}} + \frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} + \frac{\sin{\left(x \right)}}{2 \sqrt{x \sin{\left(x \right)} + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

sin(x)   x*cos(x)               
------ + --------               
  2         2          sin(x)   
----------------- + ------------
   ______________       ________
 \/ 1 + x*sin(x)    2*\/ cos(x)

$$\frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} + \frac{\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}}{\sqrt{x \sin{\left(x \right)} + 1}}$$

Simplify

The second derivative [src]

                   2                          2                           
    ________    sin (x)    (x*cos(x) + sin(x))    2*(-2*cos(x) + x*sin(x))
2*\/ cos(x)  + --------- - -------------------- - ------------------------
                  3/2                     3/2           ______________    
               cos   (x)    (1 + x*sin(x))            \/ 1 + x*sin(x)     
--------------------------------------------------------------------------
                                    4

$$\frac{\frac{\sin^{2}{\left(x \right)}}{\cos^{\frac{3}{2}}{\left(x \right)}} + 2 \sqrt{\cos{\left(x \right)}} - \frac{2 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right)}{\sqrt{x \sin{\left(x \right)} + 1}} - \frac{\left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{2}}{\left(x \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}}{4}$$

Simplify

The third derivative [src]

                                                              3        3                                                  
  4*(3*sin(x) + x*cos(x))    2*sin(x)    3*(x*cos(x) + sin(x))    3*sin (x)   6*(-2*cos(x) + x*sin(x))*(x*cos(x) + sin(x))
- ----------------------- + ---------- + ---------------------- + --------- + --------------------------------------------
        ______________        ________                   5/2         5/2                                 3/2              
      \/ 1 + x*sin(x)       \/ cos(x)      (1 + x*sin(x))         cos   (x)                (1 + x*sin(x))                 
--------------------------------------------------------------------------------------------------------------------------
                                                            8

$$\frac{\frac{3 \sin^{3}{\left(x \right)}}{\cos^{\frac{5}{2}}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{\sqrt{\cos{\left(x \right)}}} - \frac{4 \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right)}{\sqrt{x \sin{\left(x \right)} + 1}} + \frac{6 \left(x \sin{\left(x \right)} - 2 \cos{\left(x \right)}\right) \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)}{\left(x \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}} + \frac{3 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)^{3}}{\left(x \sin{\left(x \right)} + 1\right)^{\frac{5}{2}}}}{8}$$

Simplify

The graph

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Derivative of sqrt(1+x*sinx)-sqrt(cosx)

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The solution