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Chapter 3: Problem 24

Differentiate the function. \(L(\theta)=\frac{\sin \theta}{2}+\frac{c}{\theta}\)

Short Answer

Expert verified
The derivative is \(L'(\theta) = \frac{1}{2}\cos\theta - \frac{c}{\theta^2}\).

Step by step solution

01

Identify the Function Components

The function given is \[L(\theta)=\frac{\sin \theta}{2}+\frac{c}{\theta}\]This is a combination of two different terms: \(\frac{1}{2}\sin\theta\) and \(\frac{c}{\theta}\). Each part will be differentiated individually. Constant \(c\) is a real number that does not depend on \(\theta\).
02

Differentiate the First Term

The first term we need to differentiate is \(\frac{1}{2}\sin\theta\). Using the derivative of \(\sin\theta\), which is \(\cos\theta\), we get:\[ \frac{d}{d\theta}\left(\frac{1}{2}\sin\theta\right) = \frac{1}{2}\cos\theta \]
03

Differentiate the Second Term

The second term is \(\frac{c}{\theta}\). This can be rewritten as \(c\cdot\theta^{-1}\). The power rule for differentiation states that \(\frac{d}{d\theta}\theta^n = n\theta^{n-1}\). Applying this, we find:\[ \frac{d}{d\theta}\left(c\theta^{-1}\right) = -c\theta^{-2} = -\frac{c}{\theta^2} \]
04

Combine the Derived Terms

Now, integrate the differentiated parts to get the derivative of the full function:\[ L'(\theta) = \frac{1}{2}\cos\theta - \frac{c}{\theta^2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions are at the heart of many biological processes and phenomena.*Sine* and *cosine* are the two most fundamental trigonometric functions that we encounter.
  • **Sine Function (\(\sin\theta\)):** Represents the ratio of the length of the opposite side to the hypotenuse of a right triangle.
  • **Cosine Function (\(\cos\theta\)):** Is the ratio of the length of the adjacent side to the hypotenuse.
When we differentiate the sine function, the result is the cosine function:\[\frac{d}{d\theta}(\sin\theta) = \cos\theta\]This identity is particularly useful in calculus for modeling cycles and wave-like phenomena, which are prevalent in life sciences. Understanding how these functions work and transition through differentiation is crucial.
Power Rule
The power rule is a foundational tool in calculus used to differentiate expressions involving powers of a variable. It states:\[\frac{d}{dx}x^n = nx^{n-1}\]This rule allows us to easily differentiate polynomial functions and terms that can be rewritten to a power form. For the expression \(\frac{c}{\theta}\), we can express it as \(c\theta^{-1}\), which makes differentiation straightforward:
  • Apply the power rule: multiply by the exponent (-1) and decrease the exponent by 1.
The differentiation yields:\[\frac{d}{d\theta}(c\theta^{-1}) = -c\theta^{-2} = -\frac{c}{\theta^2}\]This transformation simplifies complex biological models into manageable mathematical forms. Remembering how to utilize the power rule will streamline these processes.
Calculus for Life Sciences
Calculus is indispensable in the life sciences, offering a powerful framework for modeling and solving biological problems. Here's how differentiation fits into this context:
  • **Growth Rates:** Calculus helps in analyzing and understanding the dynamics of population growth and decay by providing models that show population change over time.
  • **Pharmacokinetics:** Differentiation can model how a drug concentration changes in a bodily system over time鈥攁 process essential for dosage determination.
  • **Ecological Modeling:** Differentiation is pivotal in developing models that predict changes in ecosystems, interaction rates, and resource distribution.
In life sciences, calculus is not just about numbers; it's about understanding the underlying biological principles and processes. Differentiation allows for precise descriptions and predictions, aiding in the advancement of research and practice.

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Most popular questions from this chapter

Find \(d y / d x\) by implicit differentiation. \(1+x=\sin \left(x y^{2}\right)\)

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