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The derivative of the function/ (x^⅓)ln²x

Derivative of (x^⅓)ln²x

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

You have entered [src] 
3 ___    2   
\/ x *log (x)
x3log⁡(x)2\sqrt[3]{x} \log{\left(x \right)}^{2} 
x^(1/3)*log(x)^2
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)=x3f{\left(x \right)} = \sqrt[3]{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x3\sqrt[3]{x} goes to 13x23\frac{1}{3 x^{\frac{2}{3}}}

    g(x)=log⁡(x)2g{\left(x \right)} = \log{\left(x \right)}^{2}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    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:

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

    The result is: log⁡(x)23x23+2log⁡(x)x23\frac{\log{\left(x \right)}^{2}}{3 x^{\frac{2}{3}}} + \frac{2 \log{\left(x \right)}}{x^{\frac{2}{3}}}

  2. Now simplify:

    (log⁡(x)+6)log⁡(x)3x23\frac{\left(\log{\left(x \right)} + 6\right) \log{\left(x \right)}}{3 x^{\frac{2}{3}}}

The answer is:

(log⁡(x)+6)log⁡(x)3x23\frac{\left(\log{\left(x \right)} + 6\right) \log{\left(x \right)}}{3 x^{\frac{2}{3}}}

The graph
02468-8-6-4-2-1010-2525
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
              2   
2*log(x)   log (x)
-------- + -------
   2/3         2/3
  x         3*x
log⁡(x)23x23+2log⁡(x)x23\frac{\log{\left(x \right)}^{2}}{3 x^{\frac{2}{3}}} + \frac{2 \log{\left(x \right)}}{x^{\frac{2}{3}}}
Simplify
The second derivative [src] 
  /                2   \
  |    log(x)   log (x)|
2*|1 - ------ - -------|
  \      3         9   /
------------------------
           5/3          
          x
2(−log⁡(x)29−log⁡(x)3+1)x53\frac{2 \left(- \frac{\log{\left(x \right)}^{2}}{9} - \frac{\log{\left(x \right)}}{3} + 1\right)}{x^{\frac{5}{3}}}
Simplify
The third derivative [src] 
  /                   2   \
  |     log(x)   5*log (x)|
2*|-2 + ------ + ---------|
  \       3          27   /
---------------------------
             8/3           
            x
2(5log⁡(x)227+log⁡(x)3−2)x83\frac{2 \left(\frac{5 \log{\left(x \right)}^{2}}{27} + \frac{\log{\left(x \right)}}{3} - 2\right)}{x^{\frac{8}{3}}}
Simplify
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