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Derivative of:
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Derivative of 7
Identical expressions
y=(x+ three)sqrt(2x- one)/(2x+ seven)
y equally (x plus 3) square root of (2x minus 1) divide by (2x plus 7)
y equally (x plus three) square root of (2x minus one) divide by (2x plus seven)
y=(x+3)√(2x-1)/(2x+7)
y=x+3sqrt2x-1/2x+7
y=(x+3)sqrt(2x-1) divide by (2x+7)
Similar expressions
y=((x+3)*sqrt(2x-1))/(2x+7)
y=(x-3)sqrt(2x-1)/(2x+7)
y=((x+3)sqrt(2x-1))/(2x+7)
y=(x+3)sqrt(2x-1)/2x+7
y=((x+3)*sqrt(2x-1))/2x+7
y=(x+3)sqrt(2x+1)/(2x+7)
y=(x+3)sqrt(2x-1)/(2x-7)

The derivative of the function/ y=(x+3)sqrt(2x-1)/(2x+7)

Derivative of y=(x+3)sqrt(2x-1)/(2x+7)

The solution

You have entered [src]

          _________
(x + 3)*\/ 2*x - 1 
-------------------
      2*x + 7

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

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \left(x + 3\right) \sqrt{2 x - 1}$ and $g{\left(x \right)} = 2 x + 7$ .

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

$\frac{- 2 \left(x + 3\right) \sqrt{2 x - 1} + \left(2 x + 7\right) \left(\frac{x + 3}{\sqrt{2 x - 1}} + \sqrt{2 x - 1}\right)}{\left(2 x + 7\right)^{2}}$
Now simplify:

$\frac{2 x^{2} + 15 x + 20}{\sqrt{2 x - 1} \left(4 x^{2} + 28 x + 49\right)}$

The answer is:

$\frac{2 x^{2} + 15 x + 20}{\sqrt{2 x - 1} \left(4 x^{2} + 28 x + 49\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  _________      x + 3                           
\/ 2*x - 1  + -----------                        
                _________       _________        
              \/ 2*x - 1    2*\/ 2*x - 1 *(x + 3)
------------------------- - ---------------------
         2*x + 7                           2     
                                  (2*x + 7)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=(x+3)sqrt(2x-1)/(2x+7)

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