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Derivative of (x^2+49)/x
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Derivative of (x-3)/sqrt(x)
Identical expressions
(x- three)/sqrt(x^ two + five)
(x minus 3) divide by square root of (x squared plus 5)
(x minus three) divide by square root of (x to the power of two plus five)
(x-3)/√(x^2+5)
(x-3)/sqrt(x2+5)
x-3/sqrtx2+5
(x-3)/sqrt(x²+5)
(x-3)/sqrt(x to the power of 2+5)
x-3/sqrtx^2+5
(x-3) divide by sqrt(x^2+5)
Similar expressions
(x+3)/sqrt(x^2+5)
(x-3)/sqrt(x^2-5)

The derivative of the function/ (x-3)/sqrt(x^2+5)

Derivative of (x-3)/sqrt(x^2+5)

The solution

You have entered [src]

   x - 3   
-----------
   ________
  /  2     
\/  x  + 5

$$\frac{x - 3}{\sqrt{x^{2} + 5}}$$

(x - 3)/sqrt(x^2 + 5)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x - 3$ and $g{\left(x \right)} = \sqrt{x^{2} + 5}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (x-3)/sqrt(x^2+5)

The graph:

Piecewise:

The solution