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Identical expressions
five /(sqrt(two)*sqrt(x)+ three)
5 divide by ( square root of (2) multiply by square root of (x) plus 3)
five divide by ( square root of (two) multiply by square root of (x) plus three)
5/(√(2)*√(x)+3)
5/(sqrt(2)sqrt(x)+3)
5/sqrt2sqrtx+3
5 divide by (sqrt(2)*sqrt(x)+3)
Similar expressions
5/(sqrt(2)*sqrt(x)-3)

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

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

The solution

You have entered [src]

       5       
---------------
  ___   ___    
\/ 2 *\/ x  + 3

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sqrt{2} \sqrt{x} + 3$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt{2} \sqrt{x} + 3\right)$ :
  1. Differentiate $\sqrt{2} \sqrt{x} + 3$ 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: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      So, the result is: $\frac{\sqrt{2}}{2 \sqrt{x}}$
    2. The derivative of the constant $3$ is zero.
    The result is: $\frac{\sqrt{2}}{2 \sqrt{x}}$
  The result of the chain rule is:
  
  $- \frac{\sqrt{2}}{2 \sqrt{x} \left(\sqrt{2} \sqrt{x} + 3\right)^{2}}$
So, the result is: $- \frac{5 \sqrt{2}}{2 \sqrt{x} \left(\sqrt{2} \sqrt{x} + 3\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              ___         
         -5*\/ 2          
--------------------------
                         2
    ___ /  ___   ___    \ 
2*\/ x *\\/ 2 *\/ x  + 3/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    /  ___                                      ___        \
    |\/ 2             4                     4*\/ 2         |
-15*|----- + -------------------- + -----------------------|
    |  5/2    2 /      ___   ___\                         2|
    | x      x *\3 + \/ 2 *\/ x /    3/2 /      ___   ___\ |
    \                               x   *\3 + \/ 2 *\/ x / /
------------------------------------------------------------
                                       2                    
                      /      ___   ___\                     
                    8*\3 + \/ 2 *\/ x /

$$- \frac{15 \left(\frac{4}{x^{2} \left(\sqrt{2} \sqrt{x} + 3\right)} + \frac{4 \sqrt{2}}{x^{\frac{3}{2}} \left(\sqrt{2} \sqrt{x} + 3\right)^{2}} + \frac{\sqrt{2}}{x^{\frac{5}{2}}}\right)}{8 \left(\sqrt{2} \sqrt{x} + 3\right)^{2}}$$

Simplify

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

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The solution