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Homework Statement
Derivative of sin(x^2cos(x))
Homework Equations
Product rule and chain rule
The Attempt at a Solution
[cos(x^2cos(x)) * (2x)(cos(x)) + (x^2)(-sin(x))]
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The derivative of sin(x^2cos(x)) is cos(x^2cos(x)) * (2xcos(x) - x^2sin(x)), using the chain rule and product rule.
To find the derivative of sin(x^2cos(x)), we first use the chain rule to take the derivative of the outer function, which is sin(u), where u = x^2cos(x). This gives us cos(x^2cos(x)) * (2xcos(x) - x^2sin(x)).
The product rule for finding the derivative of sin(x^2cos(x)) is (f'g) + (fg'), where f = sin(x^2cos(x)) and g = x^2cos(x). This gives us cos(x^2cos(x)) * (2xcos(x) - x^2sin(x)) + x^2cos(x) * (-sin(x^2cos(x)) * 2xsin(x)).
Yes, the derivative of sin(x^2cos(x)) can be further simplified to cos(x^2cos(x)) * (2xcos(x) - x^2sin(x)) + 2x^3cos^2(x)sin(x)sin(x^2cos(x)).
To find the critical points of sin(x^2cos(x)), we first set the derivative equal to 0 and solve for x. This gives us x = 0 or x = 1. Additionally, we must check for any points where the derivative is undefined, which in this case is when cos(x) = 0. Therefore, the critical points are x = 0, x = 1, and any values of x where cos(x) = 0.