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The derivative of the function/ 2*ln(3x)*x^-1

Derivative of 2*ln(3x)*x^-1

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

You have entered [src] 
2*log(3*x)
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    x
2log⁡(3x)x\frac{2 \log{\left(3 x \right)}}{x} 
(2*log(3*x))/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)=2log⁡(3x)f{\left(x \right)} = 2 \log{\left(3 x \right)} and g(x)=xg{\left(x \right)} = x.

    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. Let u=3xu = 3 x.

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

      3. Then, apply the chain rule. Multiply by ddx3x\frac{d}{d x} 3 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: 33

        The result of the chain rule is:

        1x\frac{1}{x}

      So, the result is: 2x\frac{2}{x}

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

    1. Apply the power rule: xx goes to 11

    Now plug in to the quotient rule:

    2−2log⁡(3x)x2\frac{2 - 2 \log{\left(3 x \right)}}{x^{2}}

  2. Now simplify:

    2(1−log⁡(3x))x2\frac{2 \left(1 - \log{\left(3 x \right)}\right)}{x^{2}}

The answer is:

2(1−log⁡(3x))x2\frac{2 \left(1 - \log{\left(3 x \right)}\right)}{x^{2}}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
2    2*log(3*x)
-- - ----------
 2        2    
x        x
−2log⁡(3x)x2+2x2- \frac{2 \log{\left(3 x \right)}}{x^{2}} + \frac{2}{x^{2}}
Simplify
The second derivative [src] 
2*(-3 + 2*log(3*x))
-------------------
          3        
         x
2(2log⁡(3x)−3)x3\frac{2 \left(2 \log{\left(3 x \right)} - 3\right)}{x^{3}}
Simplify
The third derivative [src] 
2*(11 - 6*log(3*x))
-------------------
          4        
         x
2(11−6log⁡(3x))x4\frac{2 \left(11 - 6 \log{\left(3 x \right)}\right)}{x^{4}}
Simplify
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