Detail solution
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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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Let .
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The derivative of cosine is negative sine:
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Then, apply the chain rule. Multiply by :
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The derivative of sine is cosine:
The result of the chain rule is:
The result of the chain rule is:
Now simplify:
The answer is:
The first derivative
[src]
-2*cos(x)*cos(sin(x))*sin(sin(x))
$$- 2 \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$
The second derivative
[src]
/ 2 2 2 2 \
2*\cos (x)*sin (sin(x)) - cos (x)*cos (sin(x)) + cos(sin(x))*sin(x)*sin(sin(x))/
$$2 \left(\sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin^{2}{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)}\right)$$
The third derivative
[src]
/ 2 2 2 \
2*\cos(sin(x))*sin(sin(x)) - 3*sin (sin(x))*sin(x) + 3*cos (sin(x))*sin(x) + 4*cos (x)*cos(sin(x))*sin(sin(x))/*cos(x)
$$2 \left(- 3 \sin{\left(x \right)} \sin^{2}{\left(\sin{\left(x \right)} \right)} + 3 \sin{\left(x \right)} \cos^{2}{\left(\sin{\left(x \right)} \right)} + 4 \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} + \sin{\left(\sin{\left(x \right)} \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)}$$