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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)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   /   2      1\
log|tan (x) + -|
   \          4/
log⁡(tan⁡2(x)+14)\log{\left(\tan^{2}{\left(x \right)} + \frac{1}{4} \right)} 
log(tan(x)^2 + 1/4)
Detail solution

  1. Let u=tan⁡2(x)+14u = \tan^{2}{\left(x \right)} + \frac{1}{4}.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(tan⁡2(x)+14)\frac{d}{d x} \left(\tan^{2}{\left(x \right)} + \frac{1}{4}\right):

    1. Differentiate tan⁡2(x)+14\tan^{2}{\left(x \right)} + \frac{1}{4} term by term:

      1. Let u=tan⁡(x)u = \tan{\left(x \right)}.

      2. Apply the power rule: u2u^{2} goes to 2u2 u

      3. Then, apply the chain rule. Multiply by ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

        1. Rewrite the function to be differentiated:

          tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

        2. Apply the quotient rule, which is:

          ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

          To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

          1. The derivative of sine is cosine:

            ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

          To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

          1. The derivative of cosine is negative sine:

            ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

          Now plug in to the quotient rule:

          sin⁡2(x)+cos⁡2(x)cos⁡2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

        The result of the chain rule is:

        2(sin⁡2(x)+cos⁡2(x))tan⁡(x)cos⁡2(x)\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 14\frac{1}{4} is zero.

      The result is: 2(sin⁡2(x)+cos⁡2(x))tan⁡(x)cos⁡2(x)\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:

    2(sin⁡2(x)+cos⁡2(x))tan⁡(x)(tan⁡2(x)+14)cos⁡2(x)\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)}}

  4. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-100100
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
(2tan⁡2(x)+2)tan⁡(x)tan⁡2(x)+14\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)
8(tan⁡2(x)+1)(−8(tan⁡2(x)+1)tan⁡2(x)4tan⁡2(x)+1+3tan⁡2(x)+1)4tan⁡2(x)+1\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)
32(tan⁡2(x)+1)(−6(tan⁡2(x)+1)24tan⁡2(x)+1+32(tan⁡2(x)+1)2tan⁡2(x)(4tan⁡2(x)+1)2−12(tan⁡2(x)+1)tan⁡2(x)4tan⁡2(x)+1+3tan⁡2(x)+2)tan⁡(x)4tan⁡2(x)+1\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
Derivative of y=ln(tg^2x+1/4)
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