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Identical expressions
ln*(sqrt((two *x+ three)/(four *x+ five)))
ln multiply by ( square root of ((2 multiply by x plus 3) divide by (4 multiply by x plus 5)))
ln multiply by ( square root of ((two multiply by x plus three) divide by (four multiply by x plus five)))
ln*(√((2*x+3)/(4*x+5)))
ln(sqrt((2x+3)/(4x+5)))
lnsqrt2x+3/4x+5
ln*(sqrt((2*x+3) divide by (4*x+5)))
Similar expressions
ln*(sqrt((2*x-3)/(4*x+5)))
ln*(sqrt((2*x+3)/(4*x-5)))

The derivative of the function/ ln*(sqrt((2*x+3)/(4*x+5)))

Derivative of ln(sqrt((2x+3)/(4*x+5)))

The solution

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   /    _________\
   |   / 2*x + 3 |
log|  /  ------- |
   \\/   4*x + 5 /

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

log(sqrt((2*x + 3)/(4*x + 5)))

Detail solution

Let $u = \sqrt{\frac{2 x + 3}{4 x + 5}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{\frac{2 x + 3}{4 x + 5}}$ :
1. Let $u = \frac{2 x + 3}{4 x + 5}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 3}{4 x + 5}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 x + 3$ and $g{\left(x \right)} = 4 x + 5$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 x + 3$ term by term:
      1. The derivative of the constant $3$ is zero.
      2. 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: $2$
      The result is: $2$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Differentiate $4 x + 5$ term by term:
      1. The derivative of the constant $5$ is zero.
      2. 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: $4$
      The result is: $4$
    Now plug in to the quotient rule:
    
    $- \frac{2}{\left(4 x + 5\right)^{2}}$
  The result of the chain rule is:
  
  $- \frac{1}{\sqrt{\frac{2 x + 3}{4 x + 5}} \left(4 x + 5\right)^{2}}$
The result of the chain rule is:

$- \frac{\frac{1}{2 x + 3} \left(4 x + 5\right)}{\left(4 x + 5\right)^{2}}$
Now simplify:

$- \frac{1}{\left(2 x + 3\right) \left(4 x + 5\right)}$

The answer is:

$- \frac{1}{\left(2 x + 3\right) \left(4 x + 5\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/   1      2*(2*x + 3)\          
|------- - -----------|*(4*x + 5)
|4*x + 5             2|          
\           (4*x + 5) /          
---------------------------------
             2*x + 3

$$\frac{\left(4 x + 5\right) \left(- \frac{2 \left(2 x + 3\right)}{\left(4 x + 5\right)^{2}} + \frac{1}{4 x + 5}\right)}{2 x + 3}$$

Simplify

The second derivative [src]

  /     2*(3 + 2*x)\ /   1         2   \
2*|-1 + -----------|*|------- + -------|
  \       5 + 4*x  / \3 + 2*x   5 + 4*x/
----------------------------------------
                3 + 2*x

$$\frac{2 \left(\frac{2 \left(2 x + 3\right)}{4 x + 5} - 1\right) \left(\frac{2}{4 x + 5} + \frac{1}{2 x + 3}\right)}{2 x + 3}$$

Simplify

The third derivative [src]

  /     2*(3 + 2*x)\ /      1            4                 2         \
8*|-1 + -----------|*|- ---------- - ---------- - -------------------|
  \       5 + 4*x  / |           2            2   (3 + 2*x)*(5 + 4*x)|
                     \  (3 + 2*x)    (5 + 4*x)                       /
----------------------------------------------------------------------
                               3 + 2*x

$$\frac{8 \left(\frac{2 \left(2 x + 3\right)}{4 x + 5} - 1\right) \left(- \frac{4}{\left(4 x + 5\right)^{2}} - \frac{2}{\left(2 x + 3\right) \left(4 x + 5\right)} - \frac{1}{\left(2 x + 3\right)^{2}}\right)}{2 x + 3}$$

Simplify

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Derivative of ln(sqrt((2x+3)/(4*x+5)))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of ln*(sqrt((2*x+3)/(4*x+5)))

The graph:

Piecewise:

The solution

Derivative of ln(sqrt((2x+3)/(4*x+5)))