Derivative of ln(x^2+x)

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   / 2    \
log\x  + x/

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

log(x^2 + x)

Detail solution

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

$\frac{2 x + 1}{x^{2} + x}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

1 + 2*x
-------
  2    
 x  + x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

            /              2\
            |     (1 + 2*x) |
2*(1 + 2*x)*|-3 + ----------|
            \     x*(1 + x) /
-----------------------------
          2        2         
         x *(1 + x)

$$\frac{2 \left(-3 + \frac{\left(2 x + 1\right)^{2}}{x \left(x + 1\right)}\right) \left(2 x + 1\right)}{x^{2} \left(x + 1\right)^{2}}$$

Simplify

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Derivative of ln(x^2+x)

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