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The derivative of the function/ x×lnx-ln5×log5(x)

Derivative of x×lnx-ln5×log5(x)

Function f() - derivative -N order at the point
v

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from to

Piecewise:

The solution

You have entered [src] 
                  log(x)
x*log(x) - log(5)*------
                  log(5)
xlog⁡(x)−log⁡(x)log⁡(5)log⁡(5)x \log{\left(x \right)} - \frac{\log{\left(x \right)}}{\log{\left(5 \right)}} \log{\left(5 \right)} 
x*log(x) - log(5)*log(x)/log(5)
Detail solution

  1. Differentiate xlog⁡(x)−log⁡(x)log⁡(5)log⁡(5)x \log{\left(x \right)} - \frac{\log{\left(x \right)}}{\log{\left(5 \right)}} \log{\left(5 \right)} term by term:

    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)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: xx goes to 11

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

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

      The result is: log⁡(x)+1\log{\left(x \right)} + 1

    2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of a constant times a function is the constant times the derivative of the function.

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

        So, the result is: 1xlog⁡(5)\frac{1}{x \log{\left(5 \right)}}

      So, the result is: −1x- \frac{1}{x}

    The result is: log⁡(x)+1−1x\log{\left(x \right)} + 1 - \frac{1}{x}

The answer is:

log⁡(x)+1−1x\log{\left(x \right)} + 1 - \frac{1}{x}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    1         
1 - - + log(x)
    x
log⁡(x)+1−1x\log{\left(x \right)} + 1 - \frac{1}{x}
Simplify
The second derivative [src] 
    1
1 + -
    x
-----
  x
1+1xx\frac{1 + \frac{1}{x}}{x}
Simplify
The third derivative [src] 
 /    2\ 
-|1 + -| 
 \    x/ 
---------
     2   
    x
−1+2xx2- \frac{1 + \frac{2}{x}}{x^{2}}
Simplify
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