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Identical expressions
tgx/x+7log two x-2/x
tgx divide by x plus 7 logarithm of 2x minus 2 divide by x
tgx divide by x plus 7 logarithm of two x minus 2 divide by x
tgx divide by x+7log2x-2 divide by x
Similar expressions
tgx/x-7log2x-2/x
tgx/x+7log2x+2/x

The derivative of the function/ tgx/x+7log2x-2/x

Derivative of tgx/x+7log2x-2/x

The solution

You have entered [src]

tan(x)                2
------ + 7*log(2*x) - -
  x                   x

$$7 \log{\left(2 x \right)} + \frac{\tan{\left(x \right)}}{x} - \frac{2}{x}$$

d /tan(x)                2\
--|------ + 7*log(2*x) - -|
dx\  x                   x/

$$\frac{d}{d x} \left(7 \log{\left(2 x \right)} + \frac{\tan{\left(x \right)}}{x} - \frac{2}{x}\right)$$

Transform

Detail solution

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

$\frac{7 x \cos{\left(2 x \right)} + 9 x - \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)} + 2}{x^{2} \left(\cos{\left(2 x \right)} + 1\right)}$

The answer is:

$\frac{7 x \cos{\left(2 x \right)} + 9 x - \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)} + 2}{x^{2} \left(\cos{\left(2 x \right)} + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                2            
2    7   1 + tan (x)   tan(x)
-- + - + ----------- - ------
 2   x        x           2  
x                        x

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

Simplify

The second derivative [src]

             /       2   \                                    
  7   4    2*\1 + tan (x)/   2*tan(x)     /       2   \       
- - - -- - --------------- + -------- + 2*\1 + tan (x)/*tan(x)
  x    2          x              2                            
      x                         x                             
--------------------------------------------------------------
                              x

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

Simplify

The third derivative [src]

  /             2                                                    /       2   \     /       2   \       \
  |/       2   \    6    7    3*tan(x)        2    /       2   \   3*\1 + tan (x)/   3*\1 + tan (x)/*tan(x)|
2*|\1 + tan (x)/  + -- + -- - -------- + 2*tan (x)*\1 + tan (x)/ + --------------- - ----------------------|
  |                  3    2       3                                        2                   x           |
  \                 x    x       x                                        x                                /
------------------------------------------------------------------------------------------------------------
                                                     x

$$\frac{2 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{7}{x^{2}} - \frac{3 \tan{\left(x \right)}}{x^{3}} + \frac{6}{x^{3}}\right)}{x}$$

Simplify

The graph

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Derivative of tgx/x+7log2x-2/x

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