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Derivative of:
Derivative of sin(3*x)/(x+1)
Derivative of (ln(3x-4))^(1/2)
Derivative of bx^2
Derivative of (3x-5)√x
Identical expressions
(ln(3x- four))^(one / two)
(ln(3x minus 4)) to the power of (1 divide by 2)
(ln(3x minus four)) to the power of (one divide by two)
(ln(3x-4))(1/2)
ln3x-41/2
ln3x-4^1/2
(ln(3x-4))^(1 divide by 2)
Similar expressions
(ln(3x+4))^(1/2)

The derivative of the function/ (ln(3x-4))^(1/2)

Derivative of (ln(3x-4))^(1/2)

The solution

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  ______________
\/ log(3*x - 4)

$$\sqrt{\log{\left(3 x - 4 \right)}}$$

sqrt(log(3*x - 4))

Detail solution

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

$\frac{3}{2 \left(3 x - 4\right) \sqrt{\log{\left(3 x - 4 \right)}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

             3              
----------------------------
              ______________
2*(3*x - 4)*\/ log(3*x - 4)

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

Simplify

The second derivative [src]

        /          1      \    
     -9*|2 + -------------|    
        \    log(-4 + 3*x)/    
-------------------------------
            2   _______________
4*(-4 + 3*x) *\/ log(-4 + 3*x)

$$- \frac{9 \left(2 + \frac{1}{\log{\left(3 x - 4 \right)}}\right)}{4 \left(3 x - 4\right)^{2} \sqrt{\log{\left(3 x - 4 \right)}}}$$

Simplify

The third derivative [src]

   /           3                 3        \
27*|1 + --------------- + ----------------|
   |    4*log(-4 + 3*x)        2          |
   \                      8*log (-4 + 3*x)/
-------------------------------------------
                 3   _______________       
       (-4 + 3*x) *\/ log(-4 + 3*x)

$$\frac{27 \left(1 + \frac{3}{4 \log{\left(3 x - 4 \right)}} + \frac{3}{8 \log{\left(3 x - 4 \right)}^{2}}\right)}{\left(3 x - 4\right)^{3} \sqrt{\log{\left(3 x - 4 \right)}}}$$

Simplify

The graph

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Derivative of (ln(3x-4))^(1/2)

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The solution