find the derivative of the function. y = x^3cosx - 8xsinx - 8cosx
To find the derivative of the function, we will use the product rule and the chain rule.
Let's begin by finding the derivative of each term separately:
1) The derivative of x^3cos(x) can be found using the product rule:
d/dx (x^3cos(x)) = 3x^2cos(x) - x^3sin(x)
2) The derivative of -8xsin(x) can be found using the product rule:
d/dx (-8xsin(x)) = -8sin(x) - 8xcos(x)
3) The derivative of -8cos(x) is simply 8sin(x).
Now, adding up the derivatives of each term, we get the derivative of the function:
dy/dx = (3x^2cos(x) - x^3sin(x)) + (-8sin(x) - 8xcos(x)) + 8sin(x)
Simplifying the expression, we have:
dy/dx = 3x^2cos(x) - x^3sin(x) - 8xcos(x)
Therefore, the derivative of the function is dy/dx = 3x^2cos(x) - x^3sin(x) - 8xcos(x).