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The derivative of the function/ 2^(tan4*x)*cos(lgx)

Derivative of 2^(tan4*x)*cos(lgx)

Function f() - derivative -N order at the point
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You have entered [src] 
 tan(4*x)            
2        *cos(log(x))
2tan⁡(4x)cos⁡(log⁡(x))2^{\tan{\left(4 x \right)}} \cos{\left(\log{\left(x \right)} \right)} 
2^tan(4*x)*cos(log(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)=2tan⁡(4x)f{\left(x \right)} = 2^{\tan{\left(4 x \right)}}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=tan⁡(4x)u = \tan{\left(4 x \right)}.

    2. ddu2u=2ulog⁡(2)\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}

    3. Then, apply the chain rule. Multiply by ddxtan⁡(4x)\frac{d}{d x} \tan{\left(4 x \right)}:

      1. Rewrite the function to be differentiated:

        tan⁡(4x)=sin⁡(4x)cos⁡(4x)\tan{\left(4 x \right)} = \frac{\sin{\left(4 x \right)}}{\cos{\left(4 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⁡(4x)f{\left(x \right)} = \sin{\left(4 x \right)} and g(x)=cos⁡(4x)g{\left(x \right)} = \cos{\left(4 x \right)}.

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

        1. Let u=4xu = 4 x.

        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 ddx4x\frac{d}{d x} 4 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: 44

          The result of the chain rule is:

          4cos⁡(4x)4 \cos{\left(4 x \right)}

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

        1. Let u=4xu = 4 x.

        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 ddx4x\frac{d}{d x} 4 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: 44

          The result of the chain rule is:

          −4sin⁡(4x)- 4 \sin{\left(4 x \right)}

        Now plug in to the quotient rule:

        4sin⁡2(4x)+4cos⁡2(4x)cos⁡2(4x)\frac{4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}}

      The result of the chain rule is:

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

    g(x)=cos⁡(log⁡(x))g{\left(x \right)} = \cos{\left(\log{\left(x \right)} \right)}; 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}

    The result is: 2tan⁡(4x)(4sin⁡2(4x)+4cos⁡2(4x))log⁡(2)cos⁡(log⁡(x))cos⁡2(4x)−2tan⁡(4x)sin⁡(log⁡(x))x\frac{2^{\tan{\left(4 x \right)}} \left(4 \sin^{2}{\left(4 x \right)} + 4 \cos^{2}{\left(4 x \right)}\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)}}{\cos^{2}{\left(4 x \right)}} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}

  2. Now simplify:

    4⋅2tan⁡(4x)log⁡(2)cos⁡(log⁡(x))cos⁡2(4x)−2tan⁡(4x)sin⁡(log⁡(x))x\frac{4 \cdot 2^{\tan{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)}}{\cos^{2}{\left(4 x \right)}} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}

The answer is:

4⋅2tan⁡(4x)log⁡(2)cos⁡(log⁡(x))cos⁡2(4x)−2tan⁡(4x)sin⁡(log⁡(x))x\frac{4 \cdot 2^{\tan{\left(4 x \right)}} \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)}}{\cos^{2}{\left(4 x \right)}} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}

The graph
02468-8-6-4-2-1010-25000002500000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   tan(4*x)                                                             
  2        *sin(log(x))    tan(4*x) /         2     \                   
- --------------------- + 2        *\4 + 4*tan (4*x)/*cos(log(x))*log(2)
            x
2tan⁡(4x)(4tan⁡2(4x)+4)log⁡(2)cos⁡(log⁡(x))−2tan⁡(4x)sin⁡(log⁡(x))x2^{\tan{\left(4 x \right)}} \left(4 \tan^{2}{\left(4 x \right)} + 4\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)} - \frac{2^{\tan{\left(4 x \right)}} \sin{\left(\log{\left(x \right)} \right)}}{x}
Simplify
The second derivative [src] 
          /                               /       2     \                                                                                                 \
 tan(4*x) |-cos(log(x)) + sin(log(x))   8*\1 + tan (4*x)/*log(2)*sin(log(x))      /       2     \ /             /       2     \       \                   |
2        *|-------------------------- - ------------------------------------ + 16*\1 + tan (4*x)/*\2*tan(4*x) + \1 + tan (4*x)/*log(2)/*cos(log(x))*log(2)|
          |             2                                x                                                                                                |
          \            x                                                                                                                                  /
2tan⁡(4x)(16((tan⁡2(4x)+1)log⁡(2)+2tan⁡(4x))(tan⁡2(4x)+1)log⁡(2)cos⁡(log⁡(x))−8(tan⁡2(4x)+1)log⁡(2)sin⁡(log⁡(x))x+sin⁡(log⁡(x))−cos⁡(log⁡(x))x2)2^{\tan{\left(4 x \right)}} \left(16 \left(\left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} + 2 \tan{\left(4 x \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)} - \frac{8 \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{\sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}\right)
Simplify
The third derivative [src] 
          /                                    /       2     \                                                          /                                 2                                            \                         /       2     \ /             /       2     \       \                   \
 tan(4*x) |  -3*cos(log(x)) + sin(log(x))   12*\1 + tan (4*x)/*(-cos(log(x)) + sin(log(x)))*log(2)      /       2     \ |         2        /       2     \     2        /       2     \                |                      48*\1 + tan (4*x)/*\2*tan(4*x) + \1 + tan (4*x)/*log(2)/*log(2)*sin(log(x))|
2        *|- ---------------------------- + ------------------------------------------------------ + 64*\1 + tan (4*x)/*\2 + 6*tan (4*x) + \1 + tan (4*x)/ *log (2) + 6*\1 + tan (4*x)/*log(2)*tan(4*x)/*cos(log(x))*log(2) - ---------------------------------------------------------------------------|
          |                3                                           2                                                                                                                                                                                           x                                     |
          \               x                                           x                                                                                                                                                                                                                                  /
2tan⁡(4x)(64(tan⁡2(4x)+1)((tan⁡2(4x)+1)2log⁡(2)2+6(tan⁡2(4x)+1)log⁡(2)tan⁡(4x)+6tan⁡2(4x)+2)log⁡(2)cos⁡(log⁡(x))−48((tan⁡2(4x)+1)log⁡(2)+2tan⁡(4x))(tan⁡2(4x)+1)log⁡(2)sin⁡(log⁡(x))x+12(sin⁡(log⁡(x))−cos⁡(log⁡(x)))(tan⁡2(4x)+1)log⁡(2)x2−sin⁡(log⁡(x))−3cos⁡(log⁡(x))x3)2^{\tan{\left(4 x \right)}} \left(64 \left(\tan^{2}{\left(4 x \right)} + 1\right) \left(\left(\tan^{2}{\left(4 x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} + 6 \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \tan{\left(4 x \right)} + 6 \tan^{2}{\left(4 x \right)} + 2\right) \log{\left(2 \right)} \cos{\left(\log{\left(x \right)} \right)} - \frac{48 \left(\left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} + 2 \tan{\left(4 x \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)} \sin{\left(\log{\left(x \right)} \right)}}{x} + \frac{12 \left(\sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}\right) \left(\tan^{2}{\left(4 x \right)} + 1\right) \log{\left(2 \right)}}{x^{2}} - \frac{\sin{\left(\log{\left(x \right)} \right)} - 3 \cos{\left(\log{\left(x \right)} \right)}}{x^{3}}\right)
Simplify
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