Question
find the derivative of the function. y = x^3cosx - 8xsinx - 8cosx
Answers
GPT 3.5
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).
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).