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Derivative of y=sec^4(x^3)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   4/ 3\
sec \x /
$$\sec^{4}{\left(x^{3} \right)}$$
d /   4/ 3\\
--\sec \x //
dx          
$$\frac{d}{d x} \sec^{4}{\left(x^{3} \right)}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

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

    1. Rewrite the function to be differentiated:

    2. Let .

    3. Apply the power rule: goes to

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

      1. Let .

      2. The derivative of cosine is negative sine:

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

        1. Apply the power rule: goes to

        The result of the chain rule is:

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The first derivative [src]
    2    4/ 3\    / 3\
12*x *sec \x /*tan\x /
$$12 x^{2} \tan{\left(x^{3} \right)} \sec^{4}{\left(x^{3} \right)}$$
The second derivative [src]
        4/ 3\ /     / 3\      3 /       2/ 3\\       3    2/ 3\\
12*x*sec \x /*\2*tan\x / + 3*x *\1 + tan \x // + 12*x *tan \x //
$$12 x \left(3 x^{3} \left(\tan^{2}{\left(x^{3} \right)} + 1\right) + 12 x^{3} \tan^{2}{\left(x^{3} \right)} + 2 \tan{\left(x^{3} \right)}\right) \sec^{4}{\left(x^{3} \right)}$$
The third derivative [src]
      4/ 3\ /   3 /       2/ 3\\       3    2/ 3\       6    3/ 3\       6 /       2/ 3\\    / 3\      / 3\\
24*sec \x /*\9*x *\1 + tan \x // + 36*x *tan \x / + 72*x *tan \x / + 63*x *\1 + tan \x //*tan\x / + tan\x //
$$24 \cdot \left(63 x^{6} \left(\tan^{2}{\left(x^{3} \right)} + 1\right) \tan{\left(x^{3} \right)} + 72 x^{6} \tan^{3}{\left(x^{3} \right)} + 9 x^{3} \left(\tan^{2}{\left(x^{3} \right)} + 1\right) + 36 x^{3} \tan^{2}{\left(x^{3} \right)} + \tan{\left(x^{3} \right)}\right) \sec^{4}{\left(x^{3} \right)}$$