Question

Which is equivalent to d/dt (2 sint cost)?

1) 1
2) 2cost-2sint
3) 2cos²t-2sin't
4) -4cost sint

Answer 1

The derivative of d/dt (2 sin(t) cos(t)) is 2 (cos^2(t) - sin^2(t)), which is option 3 and equivalent to -4 sin(t) cos(t), using several trigonometric identities and the product rule of differentiation.

Step-by-step explanation:

To find the equivalent of d/dt (2 sin(t) cos(t)), we can use the product rule of differentiation which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Applying this rule:

d/dt (2 sin(t) cos(t)) = 2 (d/dt (sin(t)) × cos(t)) + 2 (sin(t) × d/dt (cos(t))).

Using the derivatives of trigonometric functions:

• d/dt (sin(t)) = cos(t)
• d/dt (cos(t)) = -sin(t)

We get:

d/dt (2 sin(t) cos(t)) = 2 (cos(t) × cos(t)) + 2 (sin(t) × -sin(t)) = 2 cos^2(t) - 2 sin^2(t).

Using the trigonometric identity sin^2(t) + cos^2(t) = 1, we can rewrite cos^2(t) as 1 - sin^2(t):

2 cos^2(t) - 2 sin^2(t) = 2 (1 - sin^2(t)) - 2 sin^2(t)

This simplifies to:

2 - 2 sin^2(t) - 2 sin^2(t) = 2 - 4 sin^2(t).

Using another trigonometric identity, sin(2 heta) = 2sin(θ)cos(θ), we can express the original function as sin(2t), so the differentiation would yield:

d/dt (2 sin(t) cos(t)) = d/dt (sin(2t)) = 2 cos(2t).

Finally, using yet another trigonometric identity, cos(2 heta) = cos^2(θ) - sin^2(θ), we arrive at:

2 cos^2(t) - 2 sin^2(t), which is option 3 and is equivalent to -4 sin(t) cos(t), which is -4 sin(t) cos(t).