49.3k views
1 vote
Explain 2 different ways to solve for the derivative s(θ)=200sinθcosθ

User Lehu
by
7.5k points

1 Answer

4 votes

To find the derivative of the function s we can use the two following approaches:

0. Product rule.

,

1. Using trigonometric identities.

Product rule.

We know that the product rule states that:


(d)/(dx)(fg)=f^(\prime)g+fg^(\prime)

Using this rule in this case we will have:


\begin{gathered} (ds)/(d\theta)=(d)/(d\theta)(200\sin\theta\cos\theta) \\ =200(d)/(d\theta)(\sin\theta\cos\theta) \\ =200(\cos\theta(d)/(d\theta)\sin\theta+\sin\theta(d)/(d\theta)\cos\theta) \\ =200(\cos\theta\cos\theta+\sin\theta(-\sin\theta)) \\ =200(\cos^2\theta-\sin^2\theta) \end{gathered}

Therefore, using the product rule, we have that:


(ds)/(d\theta)=200(\cos^2\theta-\sin^2\theta)

Using trigonometric identities.

We can also calculate the derivative if we remember that:


\sin2\theta=2\sin\theta\cos\theta

Then, in this case we have:


\begin{gathered} (ds)/(d\theta)=(d)/(d\theta)(200\sin\theta\cos\theta) \\ =100(d)/(d\theta)(2\sin\theta\cos\theta) \\ =100(d)/(d\theta)\sin2\theta \\ =100(2\cos2\theta) \\ =200\cos2\theta \end{gathered}

Therefore, using trigonometric identities, we have that:


(ds)/(d\theta)=200\cos2\theta

Note: Both results are equivalent, to prove it we just need to remember that:


\cos^2\theta-\sin^2\theta=\cos2\theta

If we use this identity in the first result we get the second one.

User Mike Furlender
by
8.3k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories