Mister Exam

Other calculators


3^(2*x)*cot(log(x))

Derivative of 3^(2*x)*cot(log(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
 2*x            
3   *cot(log(x))
32xcot(log(x))3^{2 x} \cot{\left(\log{\left(x \right)} \right)}
d / 2*x            \
--\3   *cot(log(x))/
dx                  
ddx32xcot(log(x))\frac{d}{d x} 3^{2 x} \cot{\left(\log{\left(x \right)} \right)}
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)=32xf{\left(x \right)} = 3^{2 x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=2xu = 2 x.

    2. ddu3u=3ulog(3)\frac{d}{d u} 3^{u} = 3^{u} \log{\left(3 \right)}

    3. Then, apply the chain rule. Multiply by ddx2x\frac{d}{d x} 2 x:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 22

      The result of the chain rule is:

      232xlog(3)2 \cdot 3^{2 x} \log{\left(3 \right)}

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

    1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

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

      2. Let u=tan(log(x))u = \tan{\left(\log{\left(x \right)} \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(log(x))\frac{d}{d x} \tan{\left(\log{\left(x \right)} \right)}:

        1. Rewrite the function to be differentiated:

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

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

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

          2. The derivative of sine is cosine:

            ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

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

            1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

            The result of the chain rule is:

            cos(log(x))x\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}

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

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

          2. The derivative of cosine is negative sine:

            dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

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

            1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

            The result of the chain rule is:

            sin(log(x))x- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}

          Now plug in to the quotient rule:

          sin2(log(x))x+cos2(log(x))xcos2(log(x))\frac{\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\cos^{2}{\left(\log{\left(x \right)} \right)}}

        The result of the chain rule is:

        sin2(log(x))x+cos2(log(x))xcos2(log(x))tan2(log(x))- \frac{\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\cos^{2}{\left(\log{\left(x \right)} \right)} \tan^{2}{\left(\log{\left(x \right)} \right)}}

      Method #2

      1. Rewrite the function to be differentiated:

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

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

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

        2. The derivative of cosine is negative sine:

          dducos(u)=sin(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

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

          1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

          The result of the chain rule is:

          sin(log(x))x- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}

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

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

        2. The derivative of sine is cosine:

          ddusin(u)=cos(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

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

          1. The derivative of log(x)\log{\left(x \right)} is 1x\frac{1}{x}.

          The result of the chain rule is:

          cos(log(x))x\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}

        Now plug in to the quotient rule:

        sin2(log(x))xcos2(log(x))xsin2(log(x))\frac{- \frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} - \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\sin^{2}{\left(\log{\left(x \right)} \right)}}

    The result is: 32x(sin2(log(x))x+cos2(log(x))x)cos2(log(x))tan2(log(x))+232xlog(3)cot(log(x))- \frac{3^{2 x} \left(\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}\right)}{\cos^{2}{\left(\log{\left(x \right)} \right)} \tan^{2}{\left(\log{\left(x \right)} \right)}} + 2 \cdot 3^{2 x} \log{\left(3 \right)} \cot{\left(\log{\left(x \right)} \right)}

  2. Now simplify:

    9x(2xlog(3)sin(2log(x))2)x(1cos(2log(x)))\frac{9^{x} \left(2 x \log{\left(3 \right)} \sin{\left(2 \log{\left(x \right)} \right)} - 2\right)}{x \left(1 - \cos{\left(2 \log{\left(x \right)} \right)}\right)}


The answer is:

9x(2xlog(3)sin(2log(x))2)x(1cos(2log(x)))\frac{9^{x} \left(2 x \log{\left(3 \right)} \sin{\left(2 \log{\left(x \right)} \right)} - 2\right)}{x \left(1 - \cos{\left(2 \log{\left(x \right)} \right)}\right)}

The graph
02468-8-6-4-2-1010-1000000000010000000000
The first derivative [src]
 2*x /        2        \                            
3   *\-1 - cot (log(x))/      2*x                   
------------------------ + 2*3   *cot(log(x))*log(3)
           x                                        
232xlog(3)cot(log(x))+32x(cot2(log(x))1)x2 \cdot 3^{2 x} \log{\left(3 \right)} \cot{\left(\log{\left(x \right)} \right)} + \frac{3^{2 x} \left(- \cot^{2}{\left(\log{\left(x \right)} \right)} - 1\right)}{x}
The second derivative [src]
     /                        /       2        \                         /       2        \       \
 2*x |     2                  \1 + cot (log(x))/*(1 + 2*cot(log(x)))   4*\1 + cot (log(x))/*log(3)|
3   *|4*log (3)*cot(log(x)) + -------------------------------------- - ---------------------------|
     |                                           2                                  x             |
     \                                          x                                                 /
32x(4log(3)2cot(log(x))4(cot2(log(x))+1)log(3)x+(2cot(log(x))+1)(cot2(log(x))+1)x2)3^{2 x} \left(4 \log{\left(3 \right)}^{2} \cot{\left(\log{\left(x \right)} \right)} - \frac{4 \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(3 \right)}}{x} + \frac{\left(2 \cot{\left(\log{\left(x \right)} \right)} + 1\right) \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right)}{x^{2}}\right)
The third derivative [src]
       /                        /       2        \ /         2                        \        2    /       2        \     /       2        \                           \
   2*x |     3                  \1 + cot (log(x))/*\2 + 3*cot (log(x)) + 3*cot(log(x))/   6*log (3)*\1 + cot (log(x))/   3*\1 + cot (log(x))/*(1 + 2*cot(log(x)))*log(3)|
2*3   *|4*log (3)*cot(log(x)) - ------------------------------------------------------- - ---------------------------- + -----------------------------------------------|
       |                                                    3                                          x                                         2                      |
       \                                                   x                                                                                    x                       /
232x(4log(3)3cot(log(x))6(cot2(log(x))+1)log(3)2x+3(2cot(log(x))+1)(cot2(log(x))+1)log(3)x2(cot2(log(x))+1)(3cot2(log(x))+3cot(log(x))+2)x3)2 \cdot 3^{2 x} \left(4 \log{\left(3 \right)}^{3} \cot{\left(\log{\left(x \right)} \right)} - \frac{6 \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(3 \right)}^{2}}{x} + \frac{3 \cdot \left(2 \cot{\left(\log{\left(x \right)} \right)} + 1\right) \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \log{\left(3 \right)}}{x^{2}} - \frac{\left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \left(3 \cot^{2}{\left(\log{\left(x \right)} \right)} + 3 \cot{\left(\log{\left(x \right)} \right)} + 2\right)}{x^{3}}\right)
The graph
Derivative of 3^(2*x)*cot(log(x))