Derivative of y=x^3log2(x)

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 3 log(x)
x *------
   log(2)

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

x^3*(log(x)/log(2))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{3} \log{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(2 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $3 x^{2} \log{\left(x \right)} + x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(2 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{3 x^{2} \log{\left(x \right)} + x^{2}}{\log{\left(2 \right)}}$
Now simplify:

$\frac{x^{2} \left(3 \log{\left(x \right)} + 1\right)}{\log{\left(2 \right)}}$

The answer is:

$\frac{x^{2} \left(3 \log{\left(x \right)} + 1\right)}{\log{\left(2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2        2       
  x      3*x *log(x)
------ + -----------
log(2)      log(2)

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

Simplify

The second derivative [src]

x*(5 + 6*log(x))
----------------
     log(2)

$$\frac{x \left(6 \log{\left(x \right)} + 5\right)}{\log{\left(2 \right)}}$$

Simplify

The third derivative [src]

11 + 6*log(x)
-------------
    log(2)

$$\frac{6 \log{\left(x \right)} + 11}{\log{\left(2 \right)}}$$

Simplify

The graph