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Derivative of:
Derivative of sin(x)/cos(x)
Derivative of cos(x)^(3)
Derivative of tan(x^3)
Derivative of 1/(2*sqrt(x))
Identical expressions
one / three *ln(sqrt(3y+ two))
1 divide by 3 multiply by ln( square root of (3y plus 2))
one divide by three multiply by ln( square root of (3y plus two))
1/3*ln(√(3y+2))
1/3ln(sqrt(3y+2))
1/3lnsqrt3y+2
1 divide by 3*ln(sqrt(3y+2))
Similar expressions
1/3*ln(sqrt(3y-2))

The derivative of the function/ 1/3*ln(sqrt(3y+2))

Derivative of 1/3*ln(sqrt(3y+2))

The solution

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   /  _________\
log\\/ 3*y + 2 /
----------------
       3

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

log(sqrt(3*y + 2))/3

Detail solution

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

$\frac{1}{2 \left(3 y + 2\right)}$

The answer is:

$\frac{1}{2 \left(3 y + 2\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     1     
-----------
2*(3*y + 2)

$$\frac{1}{2 \left(3 y + 2\right)}$$

Simplify

The second derivative [src]

    -3      
------------
           2
2*(2 + 3*y)

$$- \frac{3}{2 \left(3 y + 2\right)^{2}}$$

Simplify

The third derivative [src]

    9     
----------
         3
(2 + 3*y)

$$\frac{9}{\left(3 y + 2\right)^{3}}$$

Simplify