If y=2xCos2x-Sin2x, find dy/dx when x=π/2.
To find dy/dx when x = π/2, we need to differentiate the given equation with respect to x and then substitute x = π/2.
Let's break down the given equation: y = 2xCos(2x) - Sin(2x)
Step 1: Differentiating the equation
We can differentiate each term separately using the rules of differentiation.
The first term, 2xCos(2x), can be separated into two parts: 2x and Cos(2x).
The derivative of 2x with respect to x is simply 2.
The derivative of Cos(2x) with respect to x can be found by applying the chain rule:
d(Cos(2x))/dx = -Sin(2x) * d(2x)/dx
= -2Sin(2x)
The second term, -Sin(2x), can be directly differentiated:
d(-Sin(2x))/dx = -2Cos(2x)
Now, we can differentiate each term separately:
d(2xCos(2x))/dx = 2 * d(x)/dx * Cos(2x) + 2x * d(Cos(2x))/dx
= 2 * Cos(2x) + 2x * (-2Sin(2x))
= 2Cos(2x) - 4xSin(2x)
d(-Sin(2x))/dx = -2Cos(2x)
Step 2: Substituting x = π/2
Now, we substitute x = π/2 into the expression we obtained from differentiating the equation:
dy/dx = 2Cos(2(π/2)) - 4(π/2)Sin(2(π/2))
= 2Cos(π) - 4(π/2)Sin(π)
= 2(-1) - 4(π/2)(0)
= -2 - 0
= -2
Therefore, when x = π/2, dy/dx is equal to -2.
Just use the product rule and chain rule
y=2x Cos2x-Sin2x
y' = 2cos2x - 2xsin2x*2 - 2cos2x
= -4x sin2x