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Similar expressions
2*(tan(x)*sqrt(x)+sqrt(x))

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

Derivative of 2(tan(x)sqrt(x)-sqrt(x))

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                2     /       2   \                /       2   \                                       
    3          ___ /       2   \    3*\1 + tan (x)/   3*tan(x)   6*\1 + tan (x)/*tan(x)       ___    2    /       2   \
- ------ + 4*\/ x *\1 + tan (x)/  - --------------- + -------- + ---------------------- + 8*\/ x *tan (x)*\1 + tan (x)/
     5/2                                    3/2           5/2              ___                                         
  4*x                                    2*x           4*x               \/ x

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

Simplify

The graph

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

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Derivative of 2(tan(x)sqrt(x)-sqrt(x))

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The solution

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Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution

Derivative of 2(tan(x)sqrt(x)-sqrt(x))