Derivative of y=ln(e^x+2x)

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   / x      \
log\e  + 2*x/

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

d /   / x      \\
--\log\e  + 2*x//
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

          2     
  /     x\      
  \2 + e /     x
- --------- + e 
          x     
   2*x + e      
----------------
           x    
    2*x + e

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

Simplify

The third derivative [src]

          3                     
  /     x\      /     x\  x     
2*\2 + e /    3*\2 + e /*e     x
----------- - ------------- + e 
          2             x       
/       x\       2*x + e        
\2*x + e /                      
--------------------------------
                   x            
            2*x + e

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

Simplify

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Derivative of y=ln(e^x+2x)

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