Derivative of log2(sqrt3x+4)

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   /  _____    \
log\\/ 3*x  + 4/
----------------
     log(2)

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

log(sqrt(3*x) + 4)/log(2)

Detail solution

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

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

The answer is:

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

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The first derivative [src]

             ___            
           \/ 3             
----------------------------
    ___ /  _____    \       
2*\/ x *\\/ 3*x  + 4/*log(2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /  ___                                      ___        \
  |\/ 3             3                     2*\/ 3         |
3*|----- + -------------------- + -----------------------|
  |  5/2    2 /      ___   ___\                         2|
  | x      x *\4 + \/ 3 *\/ x /    3/2 /      ___   ___\ |
  \                               x   *\4 + \/ 3 *\/ x / /
----------------------------------------------------------
                  /      ___   ___\                       
                8*\4 + \/ 3 *\/ x /*log(2)

$$\frac{3 \left(\frac{3}{x^{2} \left(\sqrt{3} \sqrt{x} + 4\right)} + \frac{2 \sqrt{3}}{x^{\frac{3}{2}} \left(\sqrt{3} \sqrt{x} + 4\right)^{2}} + \frac{\sqrt{3}}{x^{\frac{5}{2}}}\right)}{8 \left(\sqrt{3} \sqrt{x} + 4\right) \log{\left(2 \right)}}$$

Simplify

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Derivative of log2(sqrt3x+4)

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