Detail solution
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Apply the product rule:
; to find :
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Apply the product rule:
; to find :
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Apply the power rule: goes to
; to find :
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The derivative of cosine is negative sine:
The result is:
; to find :
-
The derivative of sine is cosine:
The result is:
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Now simplify:
The answer is:
The first derivative
[src]
2
x*cos (x) + (-x*sin(x) + cos(x))*sin(x)
$$x \cos^{2}{\left(x \right)} + \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)}$$
The second derivative
[src]
-((2*sin(x) + x*cos(x))*sin(x) + 2*(-cos(x) + x*sin(x))*cos(x) + x*cos(x)*sin(x))
$$- (x \sin{\left(x \right)} \cos{\left(x \right)} + 2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos{\left(x \right)} + \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)})$$
The third derivative
[src]
2
(-3*cos(x) + x*sin(x))*sin(x) - x*cos (x) - 3*(2*sin(x) + x*cos(x))*cos(x) + 3*(-cos(x) + x*sin(x))*sin(x)
$$- x \cos^{2}{\left(x \right)} + \left(x \sin{\left(x \right)} - 3 \cos{\left(x \right)}\right) \sin{\left(x \right)} + 3 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)} - 3 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \cos{\left(x \right)}$$