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ln^3(3x)
The derivative of the function/ ln^3(3x)

Derivative of ln^3(3x)

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

You have entered [src] 
   3     
log (3*x)
log⁡(3x)3\log{\left(3 x \right)}^{3} 
log(3*x)^3
Detail solution

  1. Let u=log⁡(3x)u = \log{\left(3 x \right)}.

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

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

    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}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     2     
3*log (3*x)
-----------
     x
3log⁡(3x)2x\frac{3 \log{\left(3 x \right)}^{2}}{x}
Simplify
The second derivative [src] 
3*(2 - log(3*x))*log(3*x)
-------------------------
             2           
            x
3(2−log⁡(3x))log⁡(3x)x2\frac{3 \left(2 - \log{\left(3 x \right)}\right) \log{\left(3 x \right)}}{x^{2}}
Simplify
The third derivative [src] 
  /       2                  \
6*\1 + log (3*x) - 3*log(3*x)/
------------------------------
               3              
              x
6(log⁡(3x)2−3log⁡(3x)+1)x3\frac{6 \left(\log{\left(3 x \right)}^{2} - 3 \log{\left(3 x \right)} + 1\right)}{x^{3}}
Simplify
The graph
Derivative of ln^3(3x)
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