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Derivative of:
Derivative of x^2*e^x
Derivative of x^2*cos(x)
Derivative of e^(5*x)
Derivative of x/(x^2+1)
Identical expressions
four /sqrt(4x+ one)
4 divide by square root of (4x plus 1)
four divide by square root of (4x plus one)
4/√(4x+1)
4/sqrt4x+1
4 divide by sqrt(4x+1)
Similar expressions
4/sqrt(4x-1)

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

Derivative of 4/sqrt(4x+1)

The solution

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     4     
-----------
  _________
\/ 4*x + 1

$$\frac{4}{\sqrt{4 x + 1}}$$

4/sqrt(4*x + 1)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    -8      
------------
         3/2
(4*x + 1)

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

Simplify

The second derivative [src]

     48     
------------
         5/2
(1 + 4*x)

$$\frac{48}{\left(4 x + 1\right)^{\frac{5}{2}}}$$

Simplify

The third derivative [src]

   -480     
------------
         7/2
(1 + 4*x)

$$- \frac{480}{\left(4 x + 1\right)^{\frac{7}{2}}}$$

Simplify

The graph

Derivative of 4/sqrt(4x+1)