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Identical expressions
y=ln(tg^2x+ one / four)
y equally ln(tg squared x plus 1 divide by 4)
y equally ln(tg squared x plus one divide by four)
y=ln(tg2x+1/4)
y=lntg2x+1/4
y=ln(tg²x+1/4)
y=ln(tg to the power of 2x+1/4)
y=lntg^2x+1/4
y=ln(tg^2x+1 divide by 4)
Similar expressions
y=ln(tg^2x-1/4)

The derivative of the function/ y=ln(tg^2x+1/4)

Derivative of y=ln(tg^2x+1/4)

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = \tan^{2}{\left(x \right)} + \frac{1}{4}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\tan^{2}{\left(x \right)} + \frac{1}{4}\right)$ :
1. Differentiate $\tan^{2}{\left(x \right)} + \frac{1}{4}$ term by term:
  1. Let $u = \tan{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
    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:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    The result of the chain rule is:
    
    $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  4. The derivative of the constant $\frac{1}{4}$ is zero.
  The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
The result of the chain rule is:

$\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + \frac{1}{4}\right) \cos^{2}{\left(x \right)}}$
Now simplify:

$\frac{8 \tan{\left(x \right)}}{3 \sin^{2}{\left(x \right)} + 1}$

The answer is:

$\frac{8 \tan{\left(x \right)}}{3 \sin^{2}{\left(x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                 /                               2                                              2        \       
                 |                  /       2   \          2    /       2   \      /       2   \     2   |       
   /       2   \ |         2      6*\1 + tan (x)/    12*tan (x)*\1 + tan (x)/   32*\1 + tan (x)/ *tan (x)|       
32*\1 + tan (x)/*|2 + 3*tan (x) - ---------------- - ------------------------ + -------------------------|*tan(x)
                 |                          2                      2                                2    |       
                 |                 1 + 4*tan (x)          1 + 4*tan (x)              /         2   \     |       
                 \                                                                   \1 + 4*tan (x)/     /       
-----------------------------------------------------------------------------------------------------------------
                                                           2                                                     
                                                  1 + 4*tan (x)

$$\frac{32 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{4 \tan^{2}{\left(x \right)} + 1} + \frac{32 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan^{2}{\left(x \right)}}{\left(4 \tan^{2}{\left(x \right)} + 1\right)^{2}} - \frac{12 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 1} + 3 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{4 \tan^{2}{\left(x \right)} + 1}$$

Simplify

The graph

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Derivative of y=ln(tg^2x+1/4)

The graph:

Piecewise:

The solution