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The derivative of the function/ ylny-y

Derivative of ylny-y

The solution

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y*log(y) - y

$$y \log{\left(y \right)} - y$$

d               
--(y*log(y) - y)
dy

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

Transform

Detail solution

Differentiate $y \log{\left(y \right)} - y$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d y} f{\left(y \right)} g{\left(y \right)} = f{\left(y \right)} \frac{d}{d y} g{\left(y \right)} + g{\left(y \right)} \frac{d}{d y} f{\left(y \right)}$
  
  $f{\left(y \right)} = y$ ; to find $\frac{d}{d y} f{\left(y \right)}$ :
  1. Apply the power rule: $y$ goes to $1$
  $g{\left(y \right)} = \log{\left(y \right)}$ ; to find $\frac{d}{d y} g{\left(y \right)}$ :
  1. The derivative of $\log{\left(y \right)}$ is $\frac{1}{y}$ .
  The result is: $\log{\left(y \right)} + 1$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $y$ goes to $1$
  So, the result is: $-1$
The result is: $\log{\left(y \right)}$

The answer is:

$\log{\left(y \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

log(y)

$$\log{\left(y \right)}$$

Simplify

The second derivative [src]

1
-
y

$$\frac{1}{y}$$

Simplify

The third derivative [src]

-1 
---
  2
 y

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

Simplify

The graph

Derivative of ylny-y