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Derivative of:
Derivative of x^2*sqrt(1-x^2)
Derivative of e^x*x^2
Derivative of (x+3)/(x-3)
Derivative of x^3/(x^2-4)
Identical expressions
one /(4x+ one)^(one / two)
1 divide by (4x plus 1) to the power of (1 divide by 2)
one divide by (4x plus one) to the power of (one divide by two)
1/(4x+1)(1/2)
1/4x+11/2
1/4x+1^1/2
1 divide by (4x+1)^(1 divide by 2)
Similar expressions
1/(4x-1)^(1/2)

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

Derivative of 1/(4x+1)^(1/2)

The solution

You have entered [src]

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

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

1/(sqrt(4*x + 1))

Detail solution

Let $u = \sqrt{4 x + 1}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
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.
      1. 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

     12     
------------
         5/2
(1 + 4*x)

$$\frac{12}{\left(4 x + 1\right)^{\frac{5}{2}}}$$

Simplify

The third derivative [src]

   -120     
------------
         7/2
(1 + 4*x)

$$- \frac{120}{\left(4 x + 1\right)^{\frac{7}{2}}}$$

Simplify