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Derivative of (-2)/x^3
Derivative of 1/3*x^3-4*x
Derivative of f(x)=2x-3
Derivative of y=arctgx
Identical expressions
sqrt(one +4x)*(one -cos(4x))
square root of (1 plus 4x) multiply by (1 minus co sinus of e of (4x))
square root of (one plus 4x) multiply by (one minus co sinus of e of (4x))
√(1+4x)*(1-cos(4x))
sqrt(1+4x)(1-cos(4x))
sqrt1+4x1-cos4x
Similar expressions
sqrt(1-4x)*(1-cos(4x))
sqrt(1+4x)*(1+cos(4x))

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

Derivative of sqrt(1+4x)*(1-cos(4x))

The solution

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

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

sqrt(1 + 4*x)*(1 - cos(4*x))

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt{4 x + 1}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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 the constant $1$ is zero.
    2. 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: $4$
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{2}{\sqrt{4 x + 1}}$
$g{\left(x \right)} = 1 - \cos{\left(4 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $1 - \cos{\left(4 x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 4 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$ :
      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: $4$
      The result of the chain rule is:
      
      $- 4 \sin{\left(4 x \right)}$
    So, the result is: $4 \sin{\left(4 x \right)}$
  The result is: $4 \sin{\left(4 x \right)}$
The result is: $\frac{2 \left(1 - \cos{\left(4 x \right)}\right)}{\sqrt{4 x + 1}} + 4 \sqrt{4 x + 1} \sin{\left(4 x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

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

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

Simplify

The second derivative [src]

  /-1 + cos(4*x)       _________             4*sin(4*x)\
4*|------------- + 4*\/ 1 + 4*x *cos(4*x) + -----------|
  |          3/2                              _________|
  \ (1 + 4*x)                               \/ 1 + 4*x /

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

Simplify

The third derivative [src]

  /      _________             6*sin(4*x)    3*(-1 + cos(4*x))   12*cos(4*x)\
8*|- 8*\/ 1 + 4*x *sin(4*x) - ------------ - ----------------- + -----------|
  |                                    3/2               5/2       _________|
  \                           (1 + 4*x)         (1 + 4*x)        \/ 1 + 4*x /

$$8 \left(- 8 \sqrt{4 x + 1} \sin{\left(4 x \right)} + \frac{12 \cos{\left(4 x \right)}}{\sqrt{4 x + 1}} - \frac{6 \sin{\left(4 x \right)}}{\left(4 x + 1\right)^{\frac{3}{2}}} - \frac{3 \left(\cos{\left(4 x \right)} - 1\right)}{\left(4 x + 1\right)^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of sqrt(1+4x)*(1-cos(4x))

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The solution