Question

If y = sin(x)cos(x), show that d²y/dx² = -2sin(2x)?

Answer 1

Step-by-step explanation:

`y=\sin x\cos x=(1)/(2)\sin2x\\y'=\cos2x\\y''=-2\sin2x`

Answer 2

To show that d²y/dx² = -2sin(2x), differentiate y = sin(x)cos(x) twice with respect to x. First, find the first derivative using the product rule, then differentiate the first derivative with respect to x again. The resulting equation is d²y/dx² = -2sin(2x).

Step-by-step explanation:

To show that d²y/dx² = -2sin(2x), we need to differentiate y = sin(x)cos(x) twice with respect to x.

We start by finding the first derivative of y using the product rule:

dy/dx = (cos(x))(cos(x)) + (sin(x))(-sin(x)) = cos²(x) - sin²(x) = cos(2x)

Next, we differentiate dy/dx with respect to x again:

d²y/dx² = d(cos(2x))/dx = -2sin(2x)

Therefore, d²y/dx² = -2sin(2x), which proves the given equation.