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Derivative of e
Derivative of (x+1)^2
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Identical expressions
sqrt((2x- five)/(2x+ three))*lg(4x+ seven)
square root of ((2x minus 5) divide by (2x plus 3)) multiply by lg(4x plus 7)
square root of ((2x minus five) divide by (2x plus three)) multiply by lg(4x plus seven)
√((2x-5)/(2x+3))*lg(4x+7)
sqrt((2x-5)/(2x+3))lg(4x+7)
sqrt2x-5/2x+3lg4x+7
sqrt((2x-5) divide by (2x+3))*lg(4x+7)
Similar expressions
sqrt((2x+5)/(2x+3))*lg(4x+7)
sqrt((2x-5)/(2x+3))*lg(4x-7)
sqrt((2x-5)/(2x-3))*lg(4x+7)

The derivative of the function/ sqrt((2x-5)/(2x+3))*lg(4x+7)

Derivative of sqrt((2x-5)/(2x+3))*lg(4x+7)

The solution

You have entered [src]

    _________             
   / 2*x - 5              
  /  ------- *log(4*x + 7)
\/   2*x + 3

$$\sqrt{\frac{2 x - 5}{2 x + 3}} \log{\left(4 x + 7 \right)}$$

sqrt((2*x - 5)/(2*x + 3))*log(4*x + 7)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt{\frac{2 x - 5}{2 x + 3}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{2 x - 5}{2 x + 3}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x - 5}{2 x + 3}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 x - 5$ and $g{\left(x \right)} = 2 x + 3$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 x - 5$ 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: $2$
      The result is: $2$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    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: $2$
    Now plug in to the quotient rule:
    
    $\frac{16}{\left(2 x + 3\right)^{2}}$
  The result of the chain rule is:
  
  $\frac{8}{\sqrt{\frac{2 x - 5}{2 x + 3}} \left(2 x + 3\right)^{2}}$
$g{\left(x \right)} = \log{\left(4 x + 7 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 4 x + 7$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x + 7\right)$ :
  1. Differentiate $4 x + 7$ 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: $4$
    2. The derivative of the constant $7$ is zero.
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{4}{4 x + 7}$
The result is: $\frac{4 \sqrt{\frac{2 x - 5}{2 x + 3}}}{4 x + 7} + \frac{8 \log{\left(4 x + 7 \right)}}{\sqrt{\frac{2 x - 5}{2 x + 3}} \left(2 x + 3\right)^{2}}$
Now simplify:

$\frac{4 \left(\left(2 x - 5\right) \left(2 x + 3\right) + 2 \left(4 x + 7\right) \log{\left(4 x + 7 \right)}\right)}{\sqrt{\frac{2 x - 5}{2 x + 3}} \left(2 x + 3\right)^{2} \left(4 x + 7\right)}$

The answer is:

$\frac{4 \left(\left(2 x - 5\right) \left(2 x + 3\right) + 2 \left(4 x + 7\right) \log{\left(4 x + 7 \right)}\right)}{\sqrt{\frac{2 x - 5}{2 x + 3}} \left(2 x + 3\right)^{2} \left(4 x + 7\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      _________                                              
      _________      / 2*x - 5  /   1       2*x - 5  \                       
     / 2*x - 5      /  ------- *|------- - ----------|*(2*x + 3)*log(4*x + 7)
4*  /  -------    \/   2*x + 3  |2*x + 3            2|                       
  \/   2*x + 3                  \          (2*x + 3) /                       
--------------- + -----------------------------------------------------------
    4*x + 7                                 2*x - 5

$$\frac{4 \sqrt{\frac{2 x - 5}{2 x + 3}}}{4 x + 7} + \frac{\sqrt{\frac{2 x - 5}{2 x + 3}} \left(2 x + 3\right) \left(- \frac{2 x - 5}{\left(2 x + 3\right)^{2}} + \frac{1}{2 x + 3}\right) \log{\left(4 x + 7 \right)}}{2 x - 5}$$

Simplify

The second derivative [src]

               /                                                      /                          -5 + 2*x\             \
               |                                                      |                     -1 + --------|             |
               |                  /     -5 + 2*x\     /     -5 + 2*x\ |   2          2           3 + 2*x |             |
    __________ |                8*|-1 + --------|     |-1 + --------|*|-------- + ------- + -------------|*log(7 + 4*x)|
   / -5 + 2*x  |      16          \     3 + 2*x /     \     3 + 2*x / \-5 + 2*x   3 + 2*x      -5 + 2*x  /             |
  /  -------- *|- ---------- - -------------------- + -----------------------------------------------------------------|
\/   3 + 2*x   |           2   (-5 + 2*x)*(7 + 4*x)                                -5 + 2*x                            |
               \  (7 + 4*x)                                                                                            /

$$\sqrt{\frac{2 x - 5}{2 x + 3}} \left(- \frac{16}{\left(4 x + 7\right)^{2}} + \frac{\left(\frac{2 x - 5}{2 x + 3} - 1\right) \left(\frac{2}{2 x + 3} + \frac{\frac{2 x - 5}{2 x + 3} - 1}{2 x - 5} + \frac{2}{2 x - 5}\right) \log{\left(4 x + 7 \right)}}{2 x - 5} - \frac{8 \left(\frac{2 x - 5}{2 x + 3} - 1\right)}{\left(2 x - 5\right) \left(4 x + 7\right)}\right)$$

Simplify

The third derivative [src]

               /                                                     /                                          2                                                                  \                                                                       \
               |                                                     |                           /     -5 + 2*x\      /     -5 + 2*x\                             /     -5 + 2*x\  |                                                                       |
               |                                                     |                           |-1 + --------|    6*|-1 + --------|                           6*|-1 + --------|  |                                   /                          -5 + 2*x\|
               |                                     /     -5 + 2*x\ |     8            8        \     3 + 2*x /      \     3 + 2*x /            8                \     3 + 2*x /  |                                   |                     -1 + --------||
               |                  /     -5 + 2*x\    |-1 + --------|*|----------- + ---------- + ---------------- + ----------------- + -------------------- + --------------------|*log(7 + 4*x)      /     -5 + 2*x\ |   2          2           3 + 2*x ||
    __________ |               48*|-1 + --------|    \     3 + 2*x / |          2            2               2                   2      (-5 + 2*x)*(3 + 2*x)   (-5 + 2*x)*(3 + 2*x)|                12*|-1 + --------|*|-------- + ------- + -------------||
   / -5 + 2*x  |   128            \     3 + 2*x /                    \(-5 + 2*x)    (3 + 2*x)      (-5 + 2*x)          (-5 + 2*x)                                                  /                   \     3 + 2*x / \-5 + 2*x   3 + 2*x      -5 + 2*x  /|
  /  -------- *|---------- + --------------------- - -------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------|
\/   3 + 2*x   |         3                       2                                                                     -5 + 2*x                                                                                       (-5 + 2*x)*(7 + 4*x)                 |
               \(7 + 4*x)    (-5 + 2*x)*(7 + 4*x)                                                                                                                                                                                                          /

$$\sqrt{\frac{2 x - 5}{2 x + 3}} \left(\frac{128}{\left(4 x + 7\right)^{3}} - \frac{\left(\frac{2 x - 5}{2 x + 3} - 1\right) \left(\frac{8}{\left(2 x + 3\right)^{2}} + \frac{6 \left(\frac{2 x - 5}{2 x + 3} - 1\right)}{\left(2 x - 5\right) \left(2 x + 3\right)} + \frac{8}{\left(2 x - 5\right) \left(2 x + 3\right)} + \frac{\left(\frac{2 x - 5}{2 x + 3} - 1\right)^{2}}{\left(2 x - 5\right)^{2}} + \frac{6 \left(\frac{2 x - 5}{2 x + 3} - 1\right)}{\left(2 x - 5\right)^{2}} + \frac{8}{\left(2 x - 5\right)^{2}}\right) \log{\left(4 x + 7 \right)}}{2 x - 5} + \frac{12 \left(\frac{2 x - 5}{2 x + 3} - 1\right) \left(\frac{2}{2 x + 3} + \frac{\frac{2 x - 5}{2 x + 3} - 1}{2 x - 5} + \frac{2}{2 x - 5}\right)}{\left(2 x - 5\right) \left(4 x + 7\right)} + \frac{48 \left(\frac{2 x - 5}{2 x + 3} - 1\right)}{\left(2 x - 5\right) \left(4 x + 7\right)^{2}}\right)$$

Simplify

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Derivative of sqrt((2x-5)/(2x+3))*lg(4x+7)

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The solution