Detail solution
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Apply the product rule:
; to find :
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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 sine is cosine:
The result of the chain rule is:
; to find :
-
Let .
-
Apply the power rule: goes to
-
Then, apply the chain rule. Multiply by :
-
The derivative of cosine is negative sine:
The result of the chain rule is:
The result is:
Now simplify:
The answer is:
The first derivative
[src]
3 5 5 3
- 4*cos (x)*sin (x) + 4*cos (x)*sin (x)
$$- 4 \sin^{5}{\left(x \right)} \cos^{3}{\left(x \right)} + 4 \sin^{3}{\left(x \right)} \cos^{5}{\left(x \right)}$$
The second derivative
[src]
2 2 / 2 / 2 2 \ 2 / 2 2 \ 2 2 \
4*cos (x)*sin (x)*\sin (x)*\- cos (x) + 3*sin (x)/ - cos (x)*\sin (x) - 3*cos (x)/ - 8*cos (x)*sin (x)/
$$4 \left(- \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)} + \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 8 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}$$
The third derivative
[src]
/ 4 / 2 2 \ 4 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \\
8*\- cos (x)*\- 3*cos (x) + 5*sin (x)/ - sin (x)*\- 5*cos (x) + 3*sin (x)/ + 6*cos (x)*sin (x)*\sin (x) - 3*cos (x)/ + 6*cos (x)*sin (x)*\- cos (x) + 3*sin (x)//*cos(x)*sin(x)
$$8 \left(6 \left(\sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \left(3 \sin^{2}{\left(x \right)} - 5 \cos^{2}{\left(x \right)}\right) \sin^{4}{\left(x \right)} + 6 \left(3 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} - \left(5 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right) \cos^{4}{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}$$