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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
       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
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of sine is cosine:

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate 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: goes to

          So, the result is:

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
 /        2\    /          3\ 
-\3 - 12*x /*cos\-3*x + 4*x / 
------------------------------
          2/         3\       
       sin \3*x - 4*x /       
$$- \frac{\left(3 - 12 x^{2}\right) \cos{\left(4 x^{3} - 3 x \right)}}{\sin^{2}{\left(- 4 x^{3} + 3 x \right)}}$$
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 //                                
$$\frac{3 \left(\frac{8 x \cos{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}} - 3 \left(4 x^{2} - 1\right)^{2} - \frac{6 \left(4 x^{2} - 1\right)^{2} \cos^{2}{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin^{2}{\left(x \left(4 x^{2} - 3\right) \right)}}\right)}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}}$$
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 //                                                                       
$$\frac{3 \left(- 72 x \left(4 x^{2} - 1\right) - \frac{144 x \left(4 x^{2} - 1\right) \cos^{2}{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin^{2}{\left(x \left(4 x^{2} - 3\right) \right)}} + \frac{45 \left(4 x^{2} - 1\right)^{3} \cos{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}} + \frac{54 \left(4 x^{2} - 1\right)^{3} \cos^{3}{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin^{3}{\left(x \left(4 x^{2} - 3\right) \right)}} + \frac{8 \cos{\left(x \left(4 x^{2} - 3\right) \right)}}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}}\right)}{\sin{\left(x \left(4 x^{2} - 3\right) \right)}}$$