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Derivative of sin(cos^9(x))

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   9   \
sin\cos (x)/
$$\sin{\left(\cos^{9}{\left(x \right)} \right)}$$
sin(cos(x)^9)
Detail solution
  1. Let .

  2. The derivative of sine is cosine:

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

    1. Let .

    2. Apply the power rule: goes to

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

      1. The derivative of cosine is negative sine:

      The result of the chain rule is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
      8       /   9   \       
-9*cos (x)*cos\cos (x)/*sin(x)
$$- 9 \sin{\left(x \right)} \cos^{8}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}$$
The second derivative [src]
     7    /     2       /   9   \        2       /   9   \        9       2       /   9   \\
9*cos (x)*\- cos (x)*cos\cos (x)/ + 8*sin (x)*cos\cos (x)/ - 9*cos (x)*sin (x)*sin\cos (x)//
$$9 \left(- 9 \sin^{2}{\left(x \right)} \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{9}{\left(x \right)} + 8 \sin^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}\right) \cos^{7}{\left(x \right)}$$
The third derivative [src]
     6    /        2       /   9   \         11       /   9   \         2       /   9   \         18       2       /   9   \          9       2       /   9   \\       
9*cos (x)*\- 56*sin (x)*cos\cos (x)/ - 27*cos  (x)*sin\cos (x)/ + 25*cos (x)*cos\cos (x)/ + 81*cos  (x)*sin (x)*cos\cos (x)/ + 216*cos (x)*sin (x)*sin\cos (x)//*sin(x)
$$9 \left(216 \sin^{2}{\left(x \right)} \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{9}{\left(x \right)} + 81 \sin^{2}{\left(x \right)} \cos^{18}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - 56 \sin^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - 27 \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{11}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$