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Identical expressions
sqrt(((two x- four)^2)+ one)
square root of (((2x minus 4) squared ) plus 1)
square root of (((two x minus four) squared ) plus one)
√(((2x-4)^2)+1)
sqrt(((2x-4)2)+1)
sqrt2x-42+1
sqrt(((2x-4)²)+1)
sqrt(((2x-4) to the power of 2)+1)
sqrt2x-4^2+1
Similar expressions
sqrt(((2x-4)^2)-1)
sqrt(((2x+4)^2)+1)

The derivative of the function/ sqrt(((2x-4)^2)+1)

Derivative of sqrt(((2x-4)^2)+1)

The solution

You have entered [src]

   ________________
  /          2     
\/  (2*x - 4)  + 1

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

sqrt((2*x - 4)^2 + 1)

Detail solution

Let $u = \left(2 x - 4\right)^{2} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(2 x - 4\right)^{2} + 1\right)$ :
1. Differentiate $\left(2 x - 4\right)^{2} + 1$ term by term:
  1. Let $u = 2 x - 4$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 4\right)$ :
    1. Differentiate $2 x - 4$ term by term:
      1. 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: $2$
      2. The derivative of the constant $-4$ is zero.
      The result is: $2$
    The result of the chain rule is:
    
    $8 x - 16$
  4. The derivative of the constant $1$ is zero.
  The result is: $8 x - 16$
The result of the chain rule is:

$\frac{8 x - 16}{2 \sqrt{\left(2 x - 4\right)^{2} + 1}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -8 + 4*x     
-------------------
   ________________
  /          2     
\/  (2*x - 4)  + 1

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

Simplify

The second derivative [src]

  /                2  \
  |      4*(-2 + x)   |
4*|1 - ---------------|
  |                  2|
  \    1 + 4*(-2 + x) /
-----------------------
     _________________ 
    /               2  
  \/  1 + 4*(-2 + x)

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

Simplify

The third derivative [src]

   /                 2  \         
   |       4*(-2 + x)   |         
48*|-1 + ---------------|*(-2 + x)
   |                   2|         
   \     1 + 4*(-2 + x) /         
----------------------------------
                        3/2       
       /              2\          
       \1 + 4*(-2 + x) /

$$\frac{48 \left(x - 2\right) \left(\frac{4 \left(x - 2\right)^{2}}{4 \left(x - 2\right)^{2} + 1} - 1\right)}{\left(4 \left(x - 2\right)^{2} + 1\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of sqrt(((2x-4)^2)+1)

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The solution