Question

Use the product rule and a trigonometric identity to prove the derivative of (sin^2*x) = sin(2x).

Option 1: The derivative is -2cos(2x).
Option 2: The derivative is 2cos(2x).
Option 3: The derivative is 2sin(2x).
Option 4: The derivative is 2cos(x).

Answer 1

To prove the derivative of (sin^2x) is 2cos(2x), we can use the product rule and the double-angle formula for sine.

Step-by-step explanation:

To prove the derivative of (sin^2x) is 2cos(2x), we will use the product rule and a trigonometric identity. The product rule states that the derivative of the product of two functions is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function. Let's start:

1. Start with the function (sin^2x).
2. Apply the product rule:
   Derivative of sin^2x = 2sin(x)cos(x).
3. Apply the double-angle formula for sine:
   2sin(x)cos(x) = 2sin(x)cos(x).
4. Replace sin(2x) with 2sin(x)cos(x).
   Therefore, the derivative of (sin^2x) is 2cos(2x).