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sin^ four *(2x-3sqrt(x))
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Similar expressions
sin^4*(2x+3sqrt(x))

The derivative of the function/ sin^4*(2x-3sqrt(x))

Derivative of sin^4*(2x-3sqrt(x))

The solution

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   4/          ___\
sin \2*x - 3*\/ x /

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

sin(2*x - 3*sqrt(x))^4

Detail solution

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

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

     3/          ___\ /       3   \    /           ___\
4*sin \2*x - 3*\/ x /*|2 - -------|*cos\-2*x + 3*\/ x /
                      |        ___|                    
                      \    2*\/ x /

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

Simplify

The second derivative [src]

                     /             2                                     2                             /           ___\    /           ___\\
   2/           ___\ |  /      3  \     2/           ___\     /      3  \     2/           ___\   3*cos\-2*x + 3*\/ x /*sin\-2*x + 3*\/ x /|
sin \-2*x + 3*\/ x /*|- |4 - -----| *sin \-2*x + 3*\/ x / + 3*|4 - -----| *cos \-2*x + 3*\/ x / - -----------------------------------------|
                     |  |      ___|                           |      ___|                                             3/2                  |
                     \  \    \/ x /                           \    \/ x /                                            x                     /

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

Simplify

The third derivative [src]

/                                                                                                       3/           ___\ /      3  \                                                      2/           ___\ /      3  \    /           ___\\                    
|                                                                                                  9*sin \-2*x + 3*\/ x /*|4 - -----|                                                27*cos \-2*x + 3*\/ x /*|4 - -----|*sin\-2*x + 3*\/ x /|                    
|               3                                     3                                                                   |      ___|        2/           ___\    /           ___\                           |      ___|                    |                    
|    /      3  \     3/           ___\     /      3  \     2/           ___\    /           ___\                          \    \/ x /   9*sin \-2*x + 3*\/ x /*cos\-2*x + 3*\/ x /                           \    \/ x /                    |    /           ___\
|- 3*|4 - -----| *cos \-2*x + 3*\/ x / + 5*|4 - -----| *sin \-2*x + 3*\/ x /*cos\-2*x + 3*\/ x / - ---------------------------------- + ------------------------------------------ + -------------------------------------------------------|*sin\-2*x + 3*\/ x /
|    |      ___|                           |      ___|                                                              3/2                                      5/2                                                 3/2                        |                    
\    \    \/ x /                           \    \/ x /                                                           2*x                                      2*x                                                 2*x                           /

$$\left(5 \left(4 - \frac{3}{\sqrt{x}}\right)^{3} \sin^{2}{\left(3 \sqrt{x} - 2 x \right)} \cos{\left(3 \sqrt{x} - 2 x \right)} - 3 \left(4 - \frac{3}{\sqrt{x}}\right)^{3} \cos^{3}{\left(3 \sqrt{x} - 2 x \right)} - \frac{9 \left(4 - \frac{3}{\sqrt{x}}\right) \sin^{3}{\left(3 \sqrt{x} - 2 x \right)}}{2 x^{\frac{3}{2}}} + \frac{27 \left(4 - \frac{3}{\sqrt{x}}\right) \sin{\left(3 \sqrt{x} - 2 x \right)} \cos^{2}{\left(3 \sqrt{x} - 2 x \right)}}{2 x^{\frac{3}{2}}} + \frac{9 \sin^{2}{\left(3 \sqrt{x} - 2 x \right)} \cos{\left(3 \sqrt{x} - 2 x \right)}}{2 x^{\frac{5}{2}}}\right) \sin{\left(3 \sqrt{x} - 2 x \right)}$$

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Derivative of sin^4*(2x-3sqrt(x))

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