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Identical expressions
sin^ three (sqrt(4x- one))
sinus of cubed ( square root of (4x minus 1))
sinus of to the power of three ( square root of (4x minus one))
sin^3(√(4x-1))
sin3(sqrt(4x-1))
sin3sqrt4x-1
sin³(sqrt(4x-1))
sin to the power of 3(sqrt(4x-1))
sin^3sqrt4x-1
Similar expressions
sin^3(sqrt(4x+1))

The derivative of the function/ sin^3(sqrt(4x-1))

Derivative of sin^3(sqrt(4x-1))

The solution

You have entered [src]

   3/  _________\
sin \\/ 4*x - 1 /

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

sin(sqrt(4*x - 1))^3

Detail solution

Let $u = \sin{\left(\sqrt{4 x - 1} \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\sqrt{4 x - 1} \right)}$ :
1. Let $u = \sqrt{4 x - 1}$ .
2. The derivative of sine is cosine:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
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:
  
  $\frac{2 \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}}$
The result of the chain rule is:

$\frac{6 \sin^{2}{\left(\sqrt{4 x - 1} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}}$
Now simplify:

$\frac{6 \sin^{2}{\left(\sqrt{4 x - 1} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}}$

The answer is:

$\frac{6 \sin^{2}{\left(\sqrt{4 x - 1} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}}$

The graph

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The first derivative [src]

     2/  _________\    /  _________\
6*sin \\/ 4*x - 1 /*cos\\/ 4*x - 1 /
------------------------------------
              _________             
            \/ 4*x - 1

$$\frac{6 \sin^{2}{\left(\sqrt{4 x - 1} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\sqrt{4 x - 1}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /     3/  __________\        3/  __________\        2/  __________\    /  __________\        2/  __________\    /  __________\        2/  __________\    /  __________\\
   |2*cos \\/ -1 + 4*x /   3*sin \\/ -1 + 4*x /   7*sin \\/ -1 + 4*x /*cos\\/ -1 + 4*x /   6*cos \\/ -1 + 4*x /*sin\\/ -1 + 4*x /   3*sin \\/ -1 + 4*x /*cos\\/ -1 + 4*x /|
24*|-------------------- + -------------------- - -------------------------------------- - -------------------------------------- + --------------------------------------|
   |             3/2                     2                              3/2                                       2                                       5/2             |
   \   (-1 + 4*x)              (-1 + 4*x)                     (-1 + 4*x)                                (-1 + 4*x)                              (-1 + 4*x)                /

$$24 \left(\frac{3 \sin^{3}{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{2}} - \frac{6 \sin{\left(\sqrt{4 x - 1} \right)} \cos^{2}{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{2}} - \frac{7 \sin^{2}{\left(\sqrt{4 x - 1} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{3}{2}}} + \frac{2 \cos^{3}{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{3}{2}}} + \frac{3 \sin^{2}{\left(\sqrt{4 x - 1} \right)} \cos{\left(\sqrt{4 x - 1} \right)}}{\left(4 x - 1\right)^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of sin^3(sqrt(4x-1))

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The solution