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Derivative of cos(3x)
Identical expressions
y=xtan(two √x)+7sin(one /x)
y equally x tangent of (2√x) plus 7 sinus of (1 divide by x)
y equally x tangent of (two √x) plus 7 sinus of (one divide by x)
y=xtan2√x+7sin1/x
y=xtan(2√x)+7sin(1 divide by x)
Similar expressions
y=xtan(2√x)-7sin(1/x)

The derivative of the function/ y=xtan(2√x)+7sin(1/x)

Derivative of y=xtan(2√x)+7sin(1/x)

The solution

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     /    ___\        /1\
x*tan\2*\/ x / + 7*sin|-|
                      \x/

$$x \tan{\left(2 \sqrt{x} \right)} + 7 \sin{\left(\frac{1}{x} \right)}$$

x*tan(2*sqrt(x)) + 7*sin(1/x)

Detail solution

Differentiate $x \tan{\left(2 \sqrt{x} \right)} + 7 \sin{\left(\frac{1}{x} \right)}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \tan{\left(2 \sqrt{x} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(2 \sqrt{x} \right)} = \frac{\sin{\left(2 \sqrt{x} \right)}}{\cos{\left(2 \sqrt{x} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(2 \sqrt{x} \right)}$ and $g{\left(x \right)} = \cos{\left(2 \sqrt{x} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 2 \sqrt{x}$ .
    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 $\frac{d}{d x} 2 \sqrt{x}$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
        
        So, the result is: $\frac{1}{\sqrt{x}}$
      The result of the chain rule is:
      
      $\frac{\cos{\left(2 \sqrt{x} \right)}}{\sqrt{x}}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 2 \sqrt{x}$ .
    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 $\frac{d}{d x} 2 \sqrt{x}$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
        
        So, the result is: $\frac{1}{\sqrt{x}}$
      The result of the chain rule is:
      
      $- \frac{\sin{\left(2 \sqrt{x} \right)}}{\sqrt{x}}$
    Now plug in to the quotient rule:
    
    $\frac{\frac{\sin^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}} + \frac{\cos^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}}}{\cos^{2}{\left(2 \sqrt{x} \right)}}$
  The result is: $\frac{x \left(\frac{\sin^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}} + \frac{\cos^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}}\right)}{\cos^{2}{\left(2 \sqrt{x} \right)}} + \tan{\left(2 \sqrt{x} \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{1}{x}$ .
  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 $\frac{d}{d x} \frac{1}{x}$ :
    1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
    The result of the chain rule is:
    
    $- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}}$
  So, the result is: $- \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{2}}$
The result is: $\frac{x \left(\frac{\sin^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}} + \frac{\cos^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}}\right)}{\cos^{2}{\left(2 \sqrt{x} \right)}} + \tan{\left(2 \sqrt{x} \right)} - \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{2}}$
Now simplify:

$\frac{\sqrt{x}}{\cos^{2}{\left(2 \sqrt{x} \right)}} + \frac{\sin{\left(4 \sqrt{x} \right)}}{2 \cos^{2}{\left(2 \sqrt{x} \right)}} - \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{2}}$

The answer is:

$\frac{\sqrt{x}}{\cos^{2}{\left(2 \sqrt{x} \right)}} + \frac{\sin{\left(4 \sqrt{x} \right)}}{2 \cos^{2}{\left(2 \sqrt{x} \right)}} - \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                 /1\               
                            7*cos|-|               
  ___ /       2/    ___\\        \x/      /    ___\
\/ x *\1 + tan \2*\/ x // - -------- + tan\2*\/ x /
                                2                  
                               x

$$\sqrt{x} \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) + \tan{\left(2 \sqrt{x} \right)} - \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{2}}$$

Simplify

The second derivative [src]

       /1\                                              /1\                        
  7*sin|-|                                        14*cos|-|     /       2/    ___\\
       \x/     /       2/    ___\\    /    ___\         \x/   3*\1 + tan \2*\/ x //
- -------- + 2*\1 + tan \2*\/ x //*tan\2*\/ x / + --------- + ---------------------
      4                                                3                 ___       
     x                                                x              2*\/ x

$$2 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) \tan{\left(2 \sqrt{x} \right)} + \frac{14 \cos{\left(\frac{1}{x} \right)}}{x^{3}} - \frac{7 \sin{\left(\frac{1}{x} \right)}}{x^{4}} + \frac{3 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right)}{2 \sqrt{x}}$$

Simplify

The third derivative [src]

        /1\                        2        /1\         /1\                                                                                                   
  42*cos|-|     /       2/    ___\\    7*cos|-|   42*sin|-|     /       2/    ___\\     /       2/    ___\\    /    ___\        2/    ___\ /       2/    ___\\
        \x/   2*\1 + tan \2*\/ x //         \x/         \x/   3*\1 + tan \2*\/ x //   3*\1 + tan \2*\/ x //*tan\2*\/ x /   4*tan \2*\/ x /*\1 + tan \2*\/ x //
- --------- + ---------------------- + -------- + --------- - --------------------- + ---------------------------------- + -----------------------------------
       4                ___                6           5                 3/2                          x                                     ___               
      x               \/ x                x           x               4*x                                                                 \/ x

$$\frac{3 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) \tan{\left(2 \sqrt{x} \right)}}{x} - \frac{42 \cos{\left(\frac{1}{x} \right)}}{x^{4}} + \frac{42 \sin{\left(\frac{1}{x} \right)}}{x^{5}} + \frac{7 \cos{\left(\frac{1}{x} \right)}}{x^{6}} + \frac{2 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right)^{2}}{\sqrt{x}} + \frac{4 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right) \tan^{2}{\left(2 \sqrt{x} \right)}}{\sqrt{x}} - \frac{3 \left(\tan^{2}{\left(2 \sqrt{x} \right)} + 1\right)}{4 x^{\frac{3}{2}}}$$

Simplify

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Derivative of y=xtan(2√x)+7sin(1/x)

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