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Derivative of 2^(3*x)
Derivative of -2/x
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Derivative of 1/cos(x)
Identical expressions
sqrt(log(x+ six))/(x- four)+ln(x^ two - four)
square root of ( logarithm of (x plus 6)) divide by (x minus 4) plus ln(x squared minus 4)
square root of ( logarithm of (x plus six)) divide by (x minus four) plus ln(x to the power of two minus four)
√(log(x+6))/(x-4)+ln(x^2-4)
sqrt(log(x+6))/(x-4)+ln(x2-4)
sqrtlogx+6/x-4+lnx2-4
sqrt(log(x+6))/(x-4)+ln(x²-4)
sqrt(log(x+6))/(x-4)+ln(x to the power of 2-4)
sqrtlogx+6/x-4+lnx^2-4
sqrt(log(x+6)) divide by (x-4)+ln(x^2-4)
Similar expressions
sqrt(log(x+6))/(x+4)+ln(x^2-4)
sqrt(log(x+6))/(x-4)-ln(x^2-4)
sqrt(log(x-6))/(x-4)+ln(x^2-4)
sqrt(log(x+6))/(x-4)+ln(x^2+4)

The derivative of the function/ sqrt(log(x+6))/(x-4)+ln(x^2-4)

Derivative of sqrt(log(x+6))/(x-4)+ln(x^2-4)

The solution

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  ____________              
\/ log(x + 6)       / 2    \
-------------- + log\x  - 4/
    x - 4

$$\log{\left(x^{2} - 4 \right)} + \frac{\sqrt{\log{\left(x + 6 \right)}}}{x - 4}$$

sqrt(log(x + 6))/(x - 4) + log(x^2 - 4)

Detail solution

Differentiate $\log{\left(x^{2} - 4 \right)} + \frac{\sqrt{\log{\left(x + 6 \right)}}}{x - 4}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{\log{\left(x + 6 \right)}}$ and $g{\left(x \right)} = x - 4$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \log{\left(x + 6 \right)}$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x + 6 \right)}$ :
    1. Let $u = x + 6$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 6\right)$ :
      1. Differentiate $x + 6$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        The derivative of the constant $6$ is zero.
        
        The result is: $1$
      The result of the chain rule is:
      
      $\frac{1}{x + 6}$
    The result of the chain rule is:
    
    $\frac{1}{2 \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 4$ term by term:
    1. The derivative of the constant $-4$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $\frac{\frac{x - 4}{2 \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}}} - \sqrt{\log{\left(x + 6 \right)}}}{\left(x - 4\right)^{2}}$
2. Let $u = x^{2} - 4$ .
3. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 4\right)$ :
  1. Differentiate $x^{2} - 4$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $-4$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{2 x}{x^{2} - 4}$
The result is: $\frac{2 x}{x^{2} - 4} + \frac{\frac{x - 4}{2 \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}}} - \sqrt{\log{\left(x + 6 \right)}}}{\left(x - 4\right)^{2}}$
Now simplify:

$\frac{4 x \left(x - 4\right)^{2} \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}} + \left(x^{2} - 4\right) \left(x - 2 \left(x + 6\right) \log{\left(x + 6 \right)} - 4\right)}{2 \left(x - 4\right)^{2} \left(x + 6\right) \left(x^{2} - 4\right) \sqrt{\log{\left(x + 6 \right)}}}$

The answer is:

$\frac{4 x \left(x - 4\right)^{2} \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}} + \left(x^{2} - 4\right) \left(x - 2 \left(x + 6\right) \log{\left(x + 6 \right)} - 4\right)}{2 \left(x - 4\right)^{2} \left(x + 6\right) \left(x^{2} - 4\right) \sqrt{\log{\left(x + 6 \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    ____________                                            
  \/ log(x + 6)     2*x                    1                
- -------------- + ------ + --------------------------------
            2       2                           ____________
     (x - 4)       x  - 4   2*(x - 4)*(x + 6)*\/ log(x + 6)

$$\frac{2 x}{x^{2} - 4} + \frac{1}{2 \left(x - 4\right) \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}}} - \frac{\sqrt{\log{\left(x + 6 \right)}}}{\left(x - 4\right)^{2}}$$

Simplify

The second derivative [src]

                2          ____________                                                                                                            
   2         4*x       2*\/ log(6 + x)                   1                                   1                                    1                
------- - ---------- + ---------------- - -------------------------------- - ---------------------------------- - ---------------------------------
      2            2              3               2           ____________                     2   ____________                     2    3/2       
-4 + x    /      2\       (-4 + x)        (-4 + x) *(6 + x)*\/ log(6 + x)    2*(-4 + x)*(6 + x) *\/ log(6 + x)    4*(-4 + x)*(6 + x) *log   (6 + x)
          \-4 + x /

$$- \frac{4 x^{2}}{\left(x^{2} - 4\right)^{2}} + \frac{2}{x^{2} - 4} - \frac{1}{2 \left(x - 4\right) \left(x + 6\right)^{2} \sqrt{\log{\left(x + 6 \right)}}} - \frac{1}{4 \left(x - 4\right) \left(x + 6\right)^{2} \log{\left(x + 6 \right)}^{\frac{3}{2}}} - \frac{1}{\left(x - 4\right)^{2} \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}}} + \frac{2 \sqrt{\log{\left(x + 6 \right)}}}{\left(x - 4\right)^{3}}$$

Simplify

The third derivative [src]

                   ____________         3                                                                                                                                                                                                                            
     12*x      6*\/ log(6 + x)      16*x                      1                                  3                                    3                                    3                                   3                                    3                
- ---------- - ---------------- + ---------- + -------------------------------- + -------------------------------- + ----------------------------------- + --------------------------------- + ---------------------------------- + ---------------------------------
           2              4                3                   3   ____________           3           ____________             2        2   ____________                     3    3/2                    2        2    3/2                            3    5/2       
  /      2\       (-4 + x)        /      2\    (-4 + x)*(6 + x) *\/ log(6 + x)    (-4 + x) *(6 + x)*\/ log(6 + x)    2*(-4 + x) *(6 + x) *\/ log(6 + x)    4*(-4 + x)*(6 + x) *log   (6 + x)   4*(-4 + x) *(6 + x) *log   (6 + x)   8*(-4 + x)*(6 + x) *log   (6 + x)
  \-4 + x /                       \-4 + x /

$$\frac{16 x^{3}}{\left(x^{2} - 4\right)^{3}} - \frac{12 x}{\left(x^{2} - 4\right)^{2}} + \frac{1}{\left(x - 4\right) \left(x + 6\right)^{3} \sqrt{\log{\left(x + 6 \right)}}} + \frac{3}{4 \left(x - 4\right) \left(x + 6\right)^{3} \log{\left(x + 6 \right)}^{\frac{3}{2}}} + \frac{3}{8 \left(x - 4\right) \left(x + 6\right)^{3} \log{\left(x + 6 \right)}^{\frac{5}{2}}} + \frac{3}{2 \left(x - 4\right)^{2} \left(x + 6\right)^{2} \sqrt{\log{\left(x + 6 \right)}}} + \frac{3}{4 \left(x - 4\right)^{2} \left(x + 6\right)^{2} \log{\left(x + 6 \right)}^{\frac{3}{2}}} + \frac{3}{\left(x - 4\right)^{3} \left(x + 6\right) \sqrt{\log{\left(x + 6 \right)}}} - \frac{6 \sqrt{\log{\left(x + 6 \right)}}}{\left(x - 4\right)^{4}}$$

Simplify

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Derivative of sqrt(log(x+6))/(x-4)+ln(x^2-4)

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The solution