Derivative of y=sqrtlog(x,3)

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    ________
   / log(x) 
  /  ------ 
\/   log(3)

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

sqrt(log(x)/log(3))

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

/  ________\
|\/ log(x) |
|----------|
|  ________|
\\/ log(3) /
------------
 2*x*log(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

       3           3    
1 + -------- + ---------
    4*log(x)        2   
               8*log (x)
------------------------
 3   ________   ________
x *\/ log(3) *\/ log(x)

$$\frac{1 + \frac{3}{4 \log{\left(x \right)}} + \frac{3}{8 \log{\left(x \right)}^{2}}}{x^{3} \sqrt{\log{\left(3 \right)}} \sqrt{\log{\left(x \right)}}}$$

Simplify

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Derivative of y=sqrtlog(x,3)

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