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The derivative of the function/ sqrt(2x+6)-x

Derivative of sqrt(2x+6)-x

The solution

You have entered [src]

  _________    
\/ 2*x + 6  - x

$$- x + \sqrt{2 x + 6}$$

sqrt(2*x + 6) - x

Detail solution

Differentiate $- x + \sqrt{2 x + 6}$ term by term:
1. Let $u = 2 x + 6$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x + 6\right)$ :
  1. Differentiate $2 x + 6$ 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: $2$
    2. The derivative of the constant $6$ is zero.
    The result is: $2$
  The result of the chain rule is:
  
  $\frac{1}{\sqrt{2 x + 6}}$
4. 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: $-1$
The result is: $-1 + \frac{1}{\sqrt{2 x + 6}}$
Now simplify:

$-1 + \frac{\sqrt{2}}{2 \sqrt{x + 3}}$

The answer is:

$-1 + \frac{\sqrt{2}}{2 \sqrt{x + 3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          1     
-1 + -----------
       _________
     \/ 2*x + 6

$$-1 + \frac{1}{\sqrt{2 x + 6}}$$

Simplify

The second derivative [src]

     ___    
  -\/ 2     
------------
         3/2
4*(3 + x)

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

Simplify

The third derivative [src]

      ___   
  3*\/ 2    
------------
         5/2
8*(3 + x)

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

Simplify

The graph

Derivative of sqrt(2x+6)-x