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((one /sqrtx)+ two)*tg3x
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Similar expressions
((1/sqrtx)-2)*tg3x

The derivative of the function/ ((1/sqrtx)+2)*tg3x

Derivative of ((1/sqrtx)+2)*tg3x

The solution

You have entered [src]

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

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

(1/(sqrt(x)) + 2)*tan(3*x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \left(2 \sqrt{x} + 1\right) \tan{\left(3 x \right)}$ and $g{\left(x \right)} = \sqrt{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 2 \sqrt{x} + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $2 \sqrt{x} + 1$ term by term:
    1. The derivative of the constant $1$ 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: $\frac{1}{\sqrt{x}}$
    The result is: $\frac{1}{\sqrt{x}}$
  $g{\left(x \right)} = \tan{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 3 x$ .
    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 $\frac{d}{d x} 3 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      The result of the chain rule is:
      
      $3 \cos{\left(3 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 3 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $3$
      The result of the chain rule is:
      
      $- 3 \sin{\left(3 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
  The result is: $\frac{\left(2 \sqrt{x} + 1\right) \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right)}{\cos^{2}{\left(3 x \right)}} + \frac{\tan{\left(3 x \right)}}{\sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
Now plug in to the quotient rule:

$\frac{\sqrt{x} \left(\frac{\left(2 \sqrt{x} + 1\right) \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right)}{\cos^{2}{\left(3 x \right)}} + \frac{\tan{\left(3 x \right)}}{\sqrt{x}}\right) - \frac{\left(2 \sqrt{x} + 1\right) \tan{\left(3 x \right)}}{2 \sqrt{x}}}{x}$
Now simplify:

$\frac{24 x^{\frac{3}{2}} + 12 x - \sin{\left(6 x \right)}}{2 x^{\frac{3}{2}} \left(\cos{\left(6 x \right)} + 1\right)}$

The answer is:

$\frac{24 x^{\frac{3}{2}} + 12 x - \sin{\left(6 x \right)}}{2 x^{\frac{3}{2}} \left(\cos{\left(6 x \right)} + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                 /       2     \     /       2     \                                                            \
  |  5*tan(3*x)   9*\1 + tan (3*x)/   9*\1 + tan (3*x)/*tan(3*x)      /       2     \ /         2     \ /      1  \|
3*|- ---------- + ----------------- - -------------------------- + 18*\1 + tan (3*x)/*\1 + 3*tan (3*x)/*|2 + -----||
  |       7/2              5/2                    3/2                                                   |      ___||
  \    8*x              4*x                      x                                                      \    \/ x //

$$3 \left(18 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(3 \tan^{2}{\left(3 x \right)} + 1\right) - \frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)}}{x^{\frac{3}{2}}} + \frac{9 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{4 x^{\frac{5}{2}}} - \frac{5 \tan{\left(3 x \right)}}{8 x^{\frac{7}{2}}}\right)$$

Simplify

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Derivative of ((1/sqrtx)+2)*tg3x

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