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The derivative of the function/ xlnx-xln5

Derivative of xlnx-xln5

The solution

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x*log(x) - x*log(5)

$$x \log{\left(x \right)} - x \log{\left(5 \right)}$$

d                      
--(x*log(x) - x*log(5))
dx

$$\frac{d}{d x} \left(x \log{\left(x \right)} - x \log{\left(5 \right)}\right)$$

Transform

Detail solution

Differentiate $x \log{\left(x \right)} - x \log{\left(5 \right)}$ term by term:
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: $\log{\left(x \right)} + 1$
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. Apply the power rule: $x$ goes to $1$
    So, the result is: $\log{\left(5 \right)}$
  So, the result is: $- \log{\left(5 \right)}$
The result is: $\log{\left(x \right)} - \log{\left(5 \right)} + 1$

The answer is:

$\log{\left(x \right)} - \log{\left(5 \right)} + 1$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1 - log(5) + log(x)

$$\log{\left(x \right)} - \log{\left(5 \right)} + 1$$

Simplify

The second derivative [src]

1
-
x

$$\frac{1}{x}$$

Simplify

The third derivative [src]

-1 
---
  2
 x

$$- \frac{1}{x^{2}}$$

Simplify

The graph

Derivative of xlnx-xln5