Detail solution
-
Apply the product rule:
; 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:
; to find :
-
Apply the power rule: goes to
The result is:
Now simplify:
The answer is:
The first derivative
[src]
2
cos (x) - 2*x*cos(x)*sin(x)
$$- 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}$$
The second derivative
[src]
/ / 2 2 \ \
2*\x*\sin (x) - cos (x)/ - 2*cos(x)*sin(x)/
$$2 \left(x \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right)$$
The third derivative
[src]
/ 2 2 \
2*\- 3*cos (x) + 3*sin (x) + 4*x*cos(x)*sin(x)/
$$2 \left(4 x \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{2}{\left(x \right)} - 3 \cos^{2}{\left(x \right)}\right)$$