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The derivative of the function/ ln(ctg^2)*x

Derivative of ln(ctg^2)*x

Function f() - derivative -N order at the point
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You have entered [src] 
   /   2   \  
log\cot (x)/*x
xlog⁡(cot⁡2(x))x \log{\left(\cot^{2}{\left(x \right)} \right)} 
log(cot(x)^2)*x
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=log⁡(cot⁡2(x))f{\left(x \right)} = \log{\left(\cot^{2}{\left(x \right)} \right)}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=cot⁡2(x)u = \cot^{2}{\left(x \right)}.

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

    3. Then, apply the chain rule. Multiply by ddxcot⁡2(x)\frac{d}{d x} \cot^{2}{\left(x \right)}:

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

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

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

        1. There are multiple ways to do this derivative.

          Method #1

          1. Rewrite the function to be differentiated:

            cot⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

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

          3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

          4. 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:

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

          Method #2

          1. Rewrite the function to be differentiated:

            cot⁡(x)=cos⁡(x)sin⁡(x)\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\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)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} and g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

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

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

            Now plug in to the quotient rule:

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

        The result of the chain rule is:

        −2(sin⁡2(x)+cos⁡2(x))cot⁡(x)cos⁡2(x)tan⁡2(x)- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

      The result of the chain rule is:

      −2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)cot⁡(x)- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot{\left(x \right)}}

    g(x)=xg{\left(x \right)} = x; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    The result is: −2x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)cot⁡(x)+log⁡(cot⁡2(x))- \frac{2 x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot{\left(x \right)}} + \log{\left(\cot^{2}{\left(x \right)} \right)}

  2. Now simplify:

    −4xsin⁡(2x)+log⁡(1tan⁡2(x))- \frac{4 x}{\sin{\left(2 x \right)}} + \log{\left(\frac{1}{\tan^{2}{\left(x \right)}} \right)}

The answer is:

−4xsin⁡(2x)+log⁡(1tan⁡2(x))- \frac{4 x}{\sin{\left(2 x \right)}} + \log{\left(\frac{1}{\tan^{2}{\left(x \right)}} \right)}

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  /          2   \               
x*\-2 - 2*cot (x)/      /   2   \
------------------ + log\cot (x)/
      cot(x)
x(−2cot⁡2(x)−2)cot⁡(x)+log⁡(cot⁡2(x))\frac{x \left(- 2 \cot^{2}{\left(x \right)} - 2\right)}{\cot{\left(x \right)}} + \log{\left(\cot^{2}{\left(x \right)} \right)}
Simplify
The second derivative [src] 
  /  /                             2\                  \
  |  |                /       2   \ |     /       2   \|
  |  |         2      \1 + cot (x)/ |   2*\1 + cot (x)/|
2*|x*|2 + 2*cot (x) - --------------| - ---------------|
  |  |                      2       |        cot(x)    |
  \  \                   cot (x)    /                  /
2(x(−(cot⁡2(x)+1)2cot⁡2(x)+2cot⁡2(x)+2)−2(cot⁡2(x)+1)cot⁡(x))2 \left(x \left(- \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 2 \cot^{2}{\left(x \right)} + 2\right) - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}}\right)
Simplify
The third derivative [src] 
  /                               2                     /                        2                  \\
  |                  /       2   \                      |           /       2   \      /       2   \||
  |         2      3*\1 + cot (x)/        /       2   \ |           \1 + cot (x)/    2*\1 + cot (x)/||
2*|6 + 6*cot (x) - ---------------- - 2*x*\1 + cot (x)/*|2*cot(x) + -------------- - ---------------||
  |                       2                             |                 3               cot(x)    ||
  \                    cot (x)                          \              cot (x)                      //
2(−2x(cot⁡2(x)+1)((cot⁡2(x)+1)2cot⁡3(x)−2(cot⁡2(x)+1)cot⁡(x)+2cot⁡(x))−3(cot⁡2(x)+1)2cot⁡2(x)+6cot⁡2(x)+6)2 \left(- 2 x \left(\cot^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{3}{\left(x \right)}} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} + 2 \cot{\left(x \right)}\right) - \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 6 \cot^{2}{\left(x \right)} + 6\right)
Simplify
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