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Derivative of:
Derivative of 5*x
Derivative of (1-x^2)^(1/2)
Derivative of x-5
Derivative of x^(3*x)
Sum of series:
f(x)
Integral of d{x}:
f(x)
f(x)
Identical expressions
f(x)=xlnx+ln²x+ one
f(x) equally xlnx plus ln²x plus 1
f(x) equally xlnx plus ln²x plus one
fx=xlnx+ln²x+1
Similar expressions
f(x)=xlnx-ln²x+1
f(x)=xlnx+ln²x-1

The derivative of the function/ f(x)=xlnx+ln²x+1

Derivative of f(x)=xlnx+ln²x+1

The solution

You have entered [src]

              2       
x*log(x) + log (x) + 1

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

x*log(x) + log(x)^2 + 1

Detail solution

Differentiate $\left(x \log{\left(x \right)} + \log{\left(x \right)}^{2}\right) + 1$ term by term:
1. Differentiate $x \log{\left(x \right)} + \log{\left(x \right)}^{2}$ 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. Let $u = \log{\left(x \right)}$ .
  3. Apply the power rule: $u^{2}$ goes to $2 u$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  The result is: $\log{\left(x \right)} + 1 + \frac{2 \log{\left(x \right)}}{x}$
2. The derivative of the constant $1$ is zero.
The result is: $\log{\left(x \right)} + 1 + \frac{2 \log{\left(x \right)}}{x}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     6   4*log(x)
-1 - - + --------
     x      x    
-----------------
         2       
        x

$$\frac{-1 + \frac{4 \log{\left(x \right)}}{x} - \frac{6}{x}}{x^{2}}$$

Simplify