Derivative of ln(x²+2x)

You have entered [src]

   / 2      \
log\x  + 2*x/

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

log(x^2 + 2*x)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

          /              2\
          |     4*(1 + x) |
4*(1 + x)*|-3 + ----------|
          \     x*(2 + x) /
---------------------------
         2        2        
        x *(2 + x)

$$\frac{4 \left(-3 + \frac{4 \left(x + 1\right)^{2}}{x \left(x + 2\right)}\right) \left(x + 1\right)}{x^{2} \left(x + 2\right)^{2}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln(x²+2x)

The graph:

Piecewise:

The solution