Question

Evaluate the following expression, where θ is a function of time: :d/dθ(sinθcosθ).

a)d/dθ(sinθcosθ)=cos²θ−sin²θ
b)d/dθ(sinθcosθ)=cosθ−sinθ
c)d/dθ(sinθcosθ)=−sin²θ−cos²θ
d)d/dθ(sinθcosθ)=−cosθ+sinθ"

Answer 1

The derivative of the expression sinθcosθ with respect to θ is evaluated using the product rule and is found to be cos²θ - sin²θ. A is the final answer.

Step-by-step explanation:

The expression d/dθ(sinθcosθ) requires the use of the product rule for differentiation, as it is a product of two functions of θ: sinθ and cosθ. According to the product rule, if we have two functions u(θ) and v(θ), their derivative u'v + uv' is obtained by differentiating each function separately and then applying this rule.

Let's denote u = sinθ and v = cosθ. Then, using the product rule, the derivative is:

• u' = cosθ (derivative of sinθ)
• v' = -sinθ (derivative of cosθ)

Now, applying the product rule:

• d/dθ(sinθcosθ) = u'v + uv'
• = cosθ(cosθ) + sinθ(-sinθ)
• = cos²θ - sin²θ

Therefore, the correct answer is a) d/dθ(sinθcosθ) = cos²θ - sin²θ.