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The derivative of the function/ (3x*ln(sin(x)))/4/3x

Derivative of (3x*ln(sin(x)))/4/3x

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
/3*x*log(sin(x))\  
|---------------|  
\       4       /  
-----------------*x
        3
x14⋅3xlog⁡(sin⁡(x))3x \frac{\frac{1}{4} \cdot 3 x \log{\left(\sin{\left(x \right)} \right)}}{3} 
((((3*x)*log(sin(x)))/4)/3)*x
Detail solution

  1. 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)=3x2log⁡(sin⁡(x))f{\left(x \right)} = 3 x^{2} \log{\left(\sin{\left(x \right)} \right)} and g(x)=12g{\left(x \right)} = 12.

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

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

      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)=x2f{\left(x \right)} = x^{2}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Apply the power rule: x2x^{2} goes to 2x2 x

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

        1. Let u=sin⁡(x)u = \sin{\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 ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

          The result of the chain rule is:

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

        The result is: x2cos⁡(x)sin⁡(x)+2xlog⁡(sin⁡(x))\frac{x^{2} \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 x \log{\left(\sin{\left(x \right)} \right)}

      So, the result is: 3x2cos⁡(x)sin⁡(x)+6xlog⁡(sin⁡(x))\frac{3 x^{2} \cos{\left(x \right)}}{\sin{\left(x \right)}} + 6 x \log{\left(\sin{\left(x \right)} \right)}

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

    1. The derivative of the constant 1212 is zero.

    Now plug in to the quotient rule:

    x2cos⁡(x)4sin⁡(x)+xlog⁡(sin⁡(x))2\frac{x^{2} \cos{\left(x \right)}}{4 \sin{\left(x \right)}} + \frac{x \log{\left(\sin{\left(x \right)} \right)}}{2}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-10001000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                             /3*x*log(sin(x))\
                             |---------------|
  /log(sin(x))   x*cos(x)\   \       4       /
x*|----------- + --------| + -----------------
  \     4        4*sin(x)/           3
x(xcos⁡(x)4sin⁡(x)+log⁡(sin⁡(x))4)+14⋅3xlog⁡(sin⁡(x))3x \left(\frac{x \cos{\left(x \right)}}{4 \sin{\left(x \right)}} + \frac{\log{\left(\sin{\left(x \right)} \right)}}{4}\right) + \frac{\frac{1}{4} \cdot 3 x \log{\left(\sin{\left(x \right)} \right)}}{3}
Simplify
The second derivative [src] 
                  /  /       2   \           \             
                  |  |    cos (x)|   2*cos(x)|   2*x*cos(x)
2*log(sin(x)) - x*|x*|1 + -------| - --------| + ----------
                  |  |       2   |    sin(x) |     sin(x)  
                  \  \    sin (x)/           /             
-----------------------------------------------------------
                             4
−x(x(1+cos⁡2(x)sin⁡2(x))−2cos⁡(x)sin⁡(x))+2xcos⁡(x)sin⁡(x)+2log⁡(sin⁡(x))4\frac{- x \left(x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) + \frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} + 2 \log{\left(\sin{\left(x \right)} \right)}}{4}
Simplify
The third derivative [src] 
      /       2   \                /       2   \                  
      |    cos (x)|   6*cos(x)     |    cos (x)| /     2*x*cos(x)\
- 3*x*|1 + -------| + -------- + x*|1 + -------|*|-3 + ----------|
      |       2   |    sin(x)      |       2   | \       sin(x)  /
      \    sin (x)/                \    sin (x)/                  
------------------------------------------------------------------
                                4
x(1+cos⁡2(x)sin⁡2(x))(2xcos⁡(x)sin⁡(x)−3)−3x(1+cos⁡2(x)sin⁡2(x))+6cos⁡(x)sin⁡(x)4\frac{x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \left(\frac{2 x \cos{\left(x \right)}}{\sin{\left(x \right)}} - 3\right) - 3 x \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) + \frac{6 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{4}
Simplify
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