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Similar expressions
1/(ln(2+x/3))

The derivative of the function/ 1/(ln(2-x/3))

Derivative of 1/(ln(2-x/3))

The solution

You have entered [src]

    1     
----------
   /    x\
log|2 - -|
   \    3/

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

1/log(2 - x/3)

Detail solution

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

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

$- \frac{1}{\left(x - 6\right) \log{\left(2 - \frac{x}{3} \right)}^{2}}$

The answer is:

$- \frac{1}{\left(x - 6\right) \log{\left(2 - \frac{x}{3} \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          1          
---------------------
  /    x\    2/    x\
3*|2 - -|*log |2 - -|
  \    3/     \    3/

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

Simplify

The second derivative [src]

            2        
    1 + ----------   
           /    x\   
        log|2 - -|   
           \    3/   
---------------------
        2    2/    x\
(-6 + x) *log |2 - -|
              \    3/

$$\frac{1 + \frac{2}{\log{\left(2 - \frac{x}{3} \right)}}}{\left(x - 6\right)^{2} \log{\left(2 - \frac{x}{3} \right)}^{2}}$$

Simplify

The third derivative [src]

   /        3             3     \
-2*|1 + ---------- + -----------|
   |       /    x\      2/    x\|
   |    log|2 - -|   log |2 - -||
   \       \    3/       \    3//
---------------------------------
              3    2/    x\      
      (-6 + x) *log |2 - -|      
                    \    3/

$$- \frac{2 \left(1 + \frac{3}{\log{\left(2 - \frac{x}{3} \right)}} + \frac{3}{\log{\left(2 - \frac{x}{3} \right)}^{2}}\right)}{\left(x - 6\right)^{3} \log{\left(2 - \frac{x}{3} \right)}^{2}}$$

Simplify

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Derivative of 1/(ln(2-x/3))

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The solution