Question

Find the derivative of the function. f(t) = e⁸ᵗ sin(2t) a) 8e⁸ᵗ sin(2t) + 16e⁸ᵗ cos(2t) b) 8e⁸ᵗ sin(2t) - 16e⁸ᵗ cos(2t) c) 8e⁸ᵗ cos(2t) + 16e⁸ᵗ sin(2t) d) 8e⁸ᵗ cos(2t) - 16e⁸ᵗ sin(2t)

Answer 1

The derivative of the function f(t) = e⁸ᵗ sin(2t) is found using the product rule. The correct answer is 8e⁸ᵗ sin(2t) + 16e⁸ᵗ cos(2t), utilizing the rules for deriving exponential and trigonometric functions.

Step-by-step explanation:

To find the derivative of the function f(t) = e⁸ᵗ sin(2t), we need to apply the product rule since it is a product of two different functions, e⁸ᵗ and sin(2t). The product rule states that the derivative of a product of two functions is d/dx [u(t) v(t)] = u'(t) v(t) + u(t) v'(t), where u(t) and v(t) are functions of t.

Let's set u(t) = e⁸ᵗ and v(t) = sin(2t). Then we take the derivatives separately:

• The derivative of u(t) = e⁸ᵗ with respect to t is u'(t) = 8e⁸ᵗ, using the exponential rule.
• The derivative of v(t) = sin(2t) with respect to t is v'(t) = 2 cos(2t), using the chain rule.

Applying the product rule, we get:

f'(t) = u'(t) v(t) + u(t) v'(t) = 8e⁸ᵗ sin(2t) + e⁸ᵗ • 2 cos(2t)

Finally, we combine like terms to get the derivative:

f'(t) = 8e⁸ᵗ sin(2t) + 2e⁸ᵗ cos(2t), which simplifies to 8e⁸ᵗ sin(2t) + 16e⁸ᵗ cos(2t).

Therefore, the correct answer is a) 8e⁸ᵗ sin(2t) + 16e⁸ᵗ cos(2t).