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Derivative of tan(x/2)
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Identical expressions
two /(ln(4x+ twenty-five))
2 divide by (ln(4x plus 25))
two divide by (ln(4x plus twenty minus five))
2/ln4x+25
2 divide by (ln(4x+25))
Similar expressions
2/(ln(4x-25))

The derivative of the function/ 2/(ln(4x+25))

Derivative of 2/(ln(4x+25))

The solution

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      2      
-------------
log(4*x + 25)

$$\frac{2}{\log{\left(4 x + 25 \right)}}$$

2/log(4*x + 25)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \log{\left(4 x + 25 \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(4 x + 25 \right)}$ :
  1. Let $u = 4 x + 25$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 25\right)$ :
    1. Differentiate $4 x + 25$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      2. The derivative of the constant $25$ is zero.
      The result is: $4$
    The result of the chain rule is:
    
    $\frac{4}{4 x + 25}$
  The result of the chain rule is:
  
  $- \frac{4}{\left(4 x + 25\right) \log{\left(4 x + 25 \right)}^{2}}$
So, the result is: $- \frac{8}{\left(4 x + 25\right) \log{\left(4 x + 25 \right)}^{2}}$
Now simplify:

$- \frac{8}{\left(4 x + 25\right) \log{\left(4 x + 25 \right)}^{2}}$

The answer is:

$- \frac{8}{\left(4 x + 25\right) \log{\left(4 x + 25 \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           -8            
-------------------------
              2          
(4*x + 25)*log (4*x + 25)

$$- \frac{8}{\left(4 x + 25\right) \log{\left(4 x + 25 \right)}^{2}}$$

Simplify

The second derivative [src]

     /          2      \  
  32*|1 + -------------|  
     \    log(25 + 4*x)/  
--------------------------
          2    2          
(25 + 4*x) *log (25 + 4*x)

$$\frac{32 \left(1 + \frac{2}{\log{\left(4 x + 25 \right)}}\right)}{\left(4 x + 25\right)^{2} \log{\left(4 x + 25 \right)}^{2}}$$

Simplify

The third derivative [src]

     /          3               3       \
-256*|1 + ------------- + --------------|
     |    log(25 + 4*x)      2          |
     \                    log (25 + 4*x)/
-----------------------------------------
                  3    2                 
        (25 + 4*x) *log (25 + 4*x)

$$- \frac{256 \left(1 + \frac{3}{\log{\left(4 x + 25 \right)}} + \frac{3}{\log{\left(4 x + 25 \right)}^{2}}\right)}{\left(4 x + 25\right)^{3} \log{\left(4 x + 25 \right)}^{2}}$$

Simplify

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Derivative of 2/(ln(4x+25))

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The solution