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

Derivative of 4sqrt(4x+1)

The solution

You have entered [src]

    _________
4*\/ 4*x + 1

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

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 $u = 4 x + 1$ .
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:
  
  $\frac{2}{\sqrt{4 x + 1}}$
So, the result is: $\frac{8}{\sqrt{4 x + 1}}$
Now simplify:

$\frac{8}{\sqrt{4 x + 1}}$

The answer is:

$\frac{8}{\sqrt{4 x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     8     
-----------
  _________
\/ 4*x + 1

$$\frac{8}{\sqrt{4 x + 1}}$$

Simplify

The second derivative [src]

    -16     
------------
         3/2
(1 + 4*x)

$$- \frac{16}{\left(4 x + 1\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

     96     
------------
         5/2
(1 + 4*x)

$$\frac{96}{\left(4 x + 1\right)^{\frac{5}{2}}}$$

Simplify