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sqrt*x(tan(x)- one)
square root of multiply by x( tangent of (x) minus 1)
square root of multiply by x( tangent of (x) minus one)
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Similar expressions
sqrt*x(tan(x)+1)

The derivative of the function/ sqrt*x(tan(x)-1)

Derivative of sqrt*x(tan(x)-1)

The solution

You have entered [src]

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

$$\sqrt{x} \left(\tan{\left(x \right)} - 1\right)$$

d /  ___             \
--\\/ x *(tan(x) - 1)/
dx

$$\frac{d}{d x} \sqrt{x} \left(\tan{\left(x \right)} - 1\right)$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = \tan{\left(x \right)} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\tan{\left(x \right)} - 1$ term by term:
  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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  3. The derivative of the constant $\left(-1\right) 1$ is zero.
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result is: $\frac{\sqrt{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)} - 1}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of sqrt*x(tan(x)-1)

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Piecewise:

The solution