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Derivative of y/2
Derivative of (x+5)/(x-1)
Derivative of x+9
Derivative of (x-4)
Identical expressions
5tgx*inx+(7x/cosx)+ nine
5tgx multiply by inx plus (7x divide by co sinus of e of x) plus 9
5tgx multiply by inx plus (7x divide by co sinus of e of x) plus nine
5tgxinx+(7x/cosx)+9
5tgxinx+7x/cosx+9
5tgx*inx+(7x divide by cosx)+9
Similar expressions
5tgx*inx+(7x/cosx)-9
5tgx*inx-(7x/cosx)+9

The derivative of the function/ 5tgx*inx+(7x/cosx)+9

Derivative of 5tgx*inx+(7x/cosx)+9

The solution

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                   7*x      
5*tan(x)*log(x) + ------ + 9
                  cos(x)

$$\left(\frac{7 x}{\cos{\left(x \right)}} + \log{\left(x \right)} 5 \tan{\left(x \right)}\right) + 9$$

(5*tan(x))*log(x) + (7*x)/cos(x) + 9

Detail solution

Differentiate $\left(\frac{7 x}{\cos{\left(x \right)}} + \log{\left(x \right)} 5 \tan{\left(x \right)}\right) + 9$ term by term:
1. Differentiate $\frac{7 x}{\cos{\left(x \right)}} + \log{\left(x \right)} 5 \tan{\left(x \right)}$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = 5 \tan{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      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 $\frac{d}{d x} f{\left(x \right)}$ :
        
        The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        To find $\frac{d}{d x} g{\left(x \right)}$ :
        
        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:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      So, the result is: $\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$
    $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result is: $\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{5 \tan{\left(x \right)}}{x}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 7 x$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    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: $7$
    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:
    
    $\frac{7 x \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{7 x \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{5 \tan{\left(x \right)}}{x}$
2. The derivative of the constant $9$ is zero.
The result is: $\frac{7 x \sin{\left(x \right)} + 7 \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{5 \tan{\left(x \right)}}{x}$
Now simplify:

$\frac{x \left(7 x \sin{\left(x \right)} + 5 \log{\left(x \right)} + 7 \cos{\left(x \right)}\right) + \frac{5 \sin{\left(2 x \right)}}{2}}{x \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{x \left(7 x \sin{\left(x \right)} + 5 \log{\left(x \right)} + 7 \cos{\left(x \right)}\right) + \frac{5 \sin{\left(2 x \right)}}{2}}{x \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  7      /         2   \          5*tan(x)   7*x*sin(x)
------ + \5 + 5*tan (x)/*log(x) + -------- + ----------
cos(x)                               x           2     
                                              cos (x)

$$\frac{7 x \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \left(5 \tan^{2}{\left(x \right)} + 5\right) \log{\left(x \right)} + \frac{7}{\cos{\left(x \right)}} + \frac{5 \tan{\left(x \right)}}{x}$$

Simplify

The second derivative [src]

                         /       2   \                                                        2   
  5*tan(x)    7*x     10*\1 + tan (x)/   14*sin(x)      /       2   \                 14*x*sin (x)
- -------- + ------ + ---------------- + --------- + 10*\1 + tan (x)/*log(x)*tan(x) + ------------
      2      cos(x)          x               2                                             3      
     x                                    cos (x)                                       cos (x)

$$\frac{14 x \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{7 x}{\cos{\left(x \right)}} + 10 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{14 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{10 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{5 \tan{\left(x \right)}}{x^{2}}$$

Simplify

The third derivative [src]

            /       2   \                               2                2                                           /       2   \                                3   
  21     15*\1 + tan (x)/   10*tan(x)      /       2   \           42*sin (x)         2    /       2   \          30*\1 + tan (x)/*tan(x)   35*x*sin(x)   42*x*sin (x)
------ - ---------------- + --------- + 10*\1 + tan (x)/ *log(x) + ---------- + 20*tan (x)*\1 + tan (x)/*log(x) + ----------------------- + ----------- + ------------
cos(x)           2               3                                     3                                                     x                   2             4      
                x               x                                   cos (x)                                                                   cos (x)       cos (x)

$$\frac{42 x \sin^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{35 x \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 10 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 20 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan^{2}{\left(x \right)} + \frac{42 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{21}{\cos{\left(x \right)}} + \frac{30 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{15 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{10 \tan{\left(x \right)}}{x^{3}}$$

Simplify

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Derivative of 5tgx*inx+(7x/cosx)+9

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