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Identical expressions
(two -sqrt(x))*tgx
(2 minus square root of (x)) multiply by tgx
(two minus square root of (x)) multiply by tgx
(2-√(x))*tgx
(2-sqrt(x))tgx
2-sqrtxtgx
Similar expressions
(2+sqrt(x))*tgx

The derivative of the function/ (2-sqrt(x))*tgx

Derivative of (2-sqrt(x))*tgx

The solution

You have entered [src]

/      ___\       
\2 - \/ x /*tan(x)

$$\left(2 - \sqrt{x}\right) \tan{\left(x \right)}$$

(2 - sqrt(x))*tan(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = 2 - \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 - \sqrt{x}$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    So, the result is: $- \frac{1}{2 \sqrt{x}}$
  The result is: $- \frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{\left(2 - \sqrt{x}\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{\tan{\left(x \right)}}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/       2   \ /      ___\    tan(x)
\1 + tan (x)/*\2 - \/ x / - -------
                                ___
                            2*\/ x

$$\left(2 - \sqrt{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan{\left(x \right)}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

         2                                                  
  1 + tan (x)   tan(x)     /       2   \ /       ___\       
- ----------- + ------ - 2*\1 + tan (x)/*\-2 + \/ x /*tan(x)
       ___         3/2                                      
     \/ x       4*x

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

Simplify

The third derivative [src]

               /       2   \     /       2   \                                                      
  3*tan(x)   3*\1 + tan (x)/   3*\1 + tan (x)/*tan(x)     /       2   \ /         2   \ /       ___\
- -------- + --------------- - ---------------------- - 2*\1 + tan (x)/*\1 + 3*tan (x)/*\-2 + \/ x /
      5/2            3/2                 ___                                                        
   8*x            4*x                  \/ x

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

Simplify

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Derivative of (2-sqrt(x))*tgx

The graph:

Piecewise:

The solution