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Identical expressions
ln*tg(2x+ one / four)
ln multiply by tg(2x plus 1 divide by 4)
ln multiply by tg(2x plus one divide by four)
lntg(2x+1/4)
lntg2x+1/4
ln*tg(2x+1 divide by 4)
Similar expressions
ln*tg(2x-1/4)

The derivative of the function/ ln*tg(2x+1/4)

Derivative of ln*tg(2x+1/4)

The solution

You have entered [src]

log(x)*tan(2*x + 1/4)

$$\log{\left(x \right)} \tan{\left(2 x + \frac{1}{4} \right)}$$

log(x)*tan(2*x + 1/4)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \tan{\left(2 x + \frac{1}{4} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(2 x + \frac{1}{4} \right)} = \frac{\sin{\left(2 x + \frac{1}{4} \right)}}{\cos{\left(2 x + \frac{1}{4} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(2 x + \frac{1}{4} \right)}$ and $g{\left(x \right)} = \cos{\left(2 x + \frac{1}{4} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2 x + \frac{1}{4}$ .
  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 $\frac{d}{d x} \left(2 x + \frac{1}{4}\right)$ :
    1. Differentiate $2 x + \frac{1}{4}$ term by term:
      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: $2$
      2. The derivative of the constant $\frac{1}{4}$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x + \frac{1}{4} \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 x + \frac{1}{4}$ .
  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 $\frac{d}{d x} \left(2 x + \frac{1}{4}\right)$ :
    1. Differentiate $2 x + \frac{1}{4}$ term by term:
      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: $2$
      2. The derivative of the constant $\frac{1}{4}$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $- 2 \sin{\left(2 x + \frac{1}{4} \right)}$
  Now plug in to the quotient rule:
  
  $\frac{2 \sin^{2}{\left(2 x + \frac{1}{4} \right)} + 2 \cos^{2}{\left(2 x + \frac{1}{4} \right)}}{\cos^{2}{\left(2 x + \frac{1}{4} \right)}}$
The result is: $\frac{\left(2 \sin^{2}{\left(2 x + \frac{1}{4} \right)} + 2 \cos^{2}{\left(2 x + \frac{1}{4} \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(2 x + \frac{1}{4} \right)}} + \frac{\tan{\left(2 x + \frac{1}{4} \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                     /       2           \                                                
  tan(1/4 + 2*x)   4*\1 + tan (1/4 + 2*x)/     /       2           \                      
- -------------- + ----------------------- + 8*\1 + tan (1/4 + 2*x)/*log(x)*tan(1/4 + 2*x)
         2                    x                                                           
        x

$$8 \left(\tan^{2}{\left(2 x + \frac{1}{4} \right)} + 1\right) \log{\left(x \right)} \tan{\left(2 x + \frac{1}{4} \right)} + \frac{4 \left(\tan^{2}{\left(2 x + \frac{1}{4} \right)} + 1\right)}{x} - \frac{\tan{\left(2 x + \frac{1}{4} \right)}}{x^{2}}$$

Simplify

The third derivative [src]

  /                   /       2           \                                                               /       2           \               \
  |tan(1/4 + 2*x)   3*\1 + tan (1/4 + 2*x)/     /       2           \ /         2           \          12*\1 + tan (1/4 + 2*x)/*tan(1/4 + 2*x)|
2*|-------------- - ----------------------- + 8*\1 + tan (1/4 + 2*x)/*\1 + 3*tan (1/4 + 2*x)/*log(x) + ---------------------------------------|
  |       3                     2                                                                                         x                   |
  \      x                     x                                                                                                              /

$$2 \left(8 \left(\tan^{2}{\left(2 x + \frac{1}{4} \right)} + 1\right) \left(3 \tan^{2}{\left(2 x + \frac{1}{4} \right)} + 1\right) \log{\left(x \right)} + \frac{12 \left(\tan^{2}{\left(2 x + \frac{1}{4} \right)} + 1\right) \tan{\left(2 x + \frac{1}{4} \right)}}{x} - \frac{3 \left(\tan^{2}{\left(2 x + \frac{1}{4} \right)} + 1\right)}{x^{2}} + \frac{\tan{\left(2 x + \frac{1}{4} \right)}}{x^{3}}\right)$$

Simplify

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Derivative of ln*tg(2x+1/4)

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Piecewise:

The solution