Detail solution
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Let .
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The derivative of sine is cosine:
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Then, apply the chain rule. Multiply by :
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Let .
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Apply the power rule: goes to
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Then, apply the chain rule. Multiply by :
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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 first derivative
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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
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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
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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)
$$9 \left(216 \sin^{2}{\left(x \right)} \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{9}{\left(x \right)} + 81 \sin^{2}{\left(x \right)} \cos^{18}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - 56 \sin^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)} - 27 \sin{\left(\cos^{9}{\left(x \right)} \right)} \cos^{11}{\left(x \right)} + 25 \cos^{2}{\left(x \right)} \cos{\left(\cos^{9}{\left(x \right)} \right)}\right) \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$