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The derivative of the function/ x*log10(x)

Derivative of x*log10(x)

The solution

You have entered [src]

   log(x)
x*-------
  log(10)

$$x \frac{\log{\left(x \right)}}{\log{\left(10 \right)}}$$

x*(log(x)/log(10))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \log{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(10 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $\log{\left(x \right)} + 1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(10 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{\log{\left(x \right)} + 1}{\log{\left(10 \right)}}$

The answer is:

$\frac{\log{\left(x \right)} + 1}{\log{\left(10 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1       log(x)
------- + -------
log(10)   log(10)

$$\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} + \frac{1}{\log{\left(10 \right)}}$$

Simplify

The second derivative [src]

    1    
---------
x*log(10)

$$\frac{1}{x \log{\left(10 \right)}}$$

Simplify

The third derivative [src]

   -1     
----------
 2        
x *log(10)

$$- \frac{1}{x^{2} \log{\left(10 \right)}}$$

Simplify

The graph

Derivative of x*log10(x)