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

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

The solution

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

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

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

Detail solution

Let $u = \sin{\left(- 4 x^{3} + 3 x \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(- 4 x^{3} + 3 x \right)}$ :
1. Let $u = - 4 x^{3} + 3 x$ .
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 $\frac{d}{d x} \left(- 4 x^{3} + 3 x\right)$ :
  1. Differentiate $- 4 x^{3} + 3 x$ 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: $3$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
      So, the result is: $- 12 x^{2}$
    The result is: $3 - 12 x^{2}$
  The result of the chain rule is:
  
  $\left(3 - 12 x^{2}\right) \cos{\left(4 x^{3} - 3 x \right)}$
The result of the chain rule is:

$- \frac{\left(3 - 12 x^{2}\right) \cos{\left(4 x^{3} - 3 x \right)}}{\sin^{2}{\left(- 4 x^{3} + 3 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /        2\    /          3\ 
-\3 - 12*x /*cos\-3*x + 4*x / 
------------------------------
          2/         3\       
       sin \3*x - 4*x /

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

Simplify

The second derivative [src]

  /                                2                                             \
  |               2     /        2\     2/  /        2\\          /  /        2\\|
  |    /        2\    6*\-1 + 4*x / *cos \x*\-3 + 4*x //   8*x*cos\x*\-3 + 4*x //|
3*|- 3*\-1 + 4*x /  - ---------------------------------- + ----------------------|
  |                             2/  /        2\\                /  /        2\\  |
  \                          sin \x*\-3 + 4*x //             sin\x*\-3 + 4*x //  /
----------------------------------------------------------------------------------
                                   /  /        2\\                                
                                sin\x*\-3 + 4*x //

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

Simplify

The third derivative [src]

  /                                                          3                                    3                                                            \
  |                          /  /        2\\      /        2\     /  /        2\\      /        2\     3/  /        2\\            2/  /        2\\ /        2\|
  |       /        2\   8*cos\x*\-3 + 4*x //   45*\-1 + 4*x / *cos\x*\-3 + 4*x //   54*\-1 + 4*x / *cos \x*\-3 + 4*x //   144*x*cos \x*\-3 + 4*x //*\-1 + 4*x /|
3*|- 72*x*\-1 + 4*x / + -------------------- + ---------------------------------- + ----------------------------------- - -------------------------------------|
  |                         /  /        2\\               /  /        2\\                      3/  /        2\\                       2/  /        2\\         |
  \                      sin\x*\-3 + 4*x //            sin\x*\-3 + 4*x //                   sin \x*\-3 + 4*x //                    sin \x*\-3 + 4*x //         /
----------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                          /  /        2\\                                                                       
                                                                       sin\x*\-3 + 4*x //

$$\frac{3 \left(- 72 x \left(4 x^{2} - 1\right) - \frac{144 x \left(4 x^{2} - 1\right) \cos^{2}{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin^{2}{\left(x \left(4 x^{2} - 3\right) \right)}} + \frac{45 \left(4 x^{2} - 1\right)^{3} \cos{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}} + \frac{54 \left(4 x^{2} - 1\right)^{3} \cos^{3}{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin^{3}{\left(x \left(4 x^{2} - 3\right) \right)}} + \frac{8 \cos{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}}\right)}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}}$$

Simplify

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

The graph:

Piecewise:

The solution