Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ y=\frac{\pi}{2} \sin \theta-\cos \theta $$

Short answer

The derivative of the function \(y=\frac{\pi}{2} \sin \theta-\cos \theta\) is \(y' = \frac{\pi}{2} \cos \theta + \sin \theta\).

Step-by-step solution

Recognize the function's components

First, it's important to recognize the individual components of the function \(y=\frac{\pi}{2} \sin \theta-\cos \theta\). Here, \(y\) is a composite of two separate terms: \(\frac{\pi}{2} \sin \theta\) and \(-\cos \theta\).

Apply the rules of differentiation to each term

Next, apply the rules of differentiation to each term separately. The derivative of \(\sin \theta\) is \(\cos \theta\) which gives \(\frac{d}{d\theta}(\frac{\pi}{2} \sin \theta) = \frac{\pi}{2} \cos \theta\). Similarly, the derivative of \(-\cos \theta\) is \(\sin \theta\), which gives \(\frac{d}{d\theta}(-\cos \theta) = \sin \theta\).

Combine the results into a derivative function

The final step is to combine the results from Step 2 into a single derivative function. Here, the derivative of the original function \(y\) is then given by \(y' = \frac{\pi}{2} \cos \theta + \sin \theta\).

Key concepts

Derivative of a Function

Understanding the derivative of a function is fundamental to calculus. The derivative measures how a function's output changes as its input changes. Essentially, it gives us the rate at which one quantity changes in relation to another. To find the derivative, we apply rules that depend on the form of the function. For example, the power rule is used for polynomials, while specific rules are designated for exponential, logarithmic, and trigonometric functions. In the given exercise, we differentiate the trigonometric functions \( \frac{\pi}{2} \sin \theta\) and \( -\cos \theta\), treating \(\pi\) as a constant. By applying the differentiation rules to these functions, students learn the procedural aspects of finding derivatives and gain insight into how functions behave at a local level.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent have specific behaviors and their derivatives follow certain patterns. For instance, the derivative of \(\sin x\) is \(\cos x\), whereas the derivative of \(\cos x\) is \( -\sin x\). These rules are essential for calculus students, especially when working with functions that involve angles and periodic behavior. By knowing these specific differentiation rules, students can solve complex problems involving trigonometric functions. In our exercise, we deal with \(\sin \theta\) and \(\cos \theta\), both common trigonometric functions often found in physics and engineering contexts, making their understanding crucial.

Chain Rule

The chain rule is a powerful technique in differentiation used when dealing with composite functions. A composite function occurs when one function is inside another; the chain rule helps us find the derivative of such functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This can be symbolically noted as \( (f(g(x)))' = f'(g(x)) \cdot g'(x)\). It is essential for students to master this rule, as it applies to a wide array of functions and is often seen in more complex calculus problems. While the chain rule isn't directly applied in our simple exercise, it is generally a stepping stone to handling more intricate functions that students will encounter later.

Differentiation Techniques

Differentiation techniques are the various methods available to tackle finding the derivative of functions. Besides basic rules and the chain rule, other techniques include the product rule, quotient rule, and implicit differentiation. The product and quotient rules are used when taking the derivative of the products and quotients of functions, respectively. Implicit differentiation is applied when the function is not explicitly solved for one variable in terms of another. Mastery of these techniques equips students to handle a broad spectrum of functions and problems in calculus, and applies to a wide range of disciplines, including economics, biology, and physics. In our initial exercise, we applied basic differentiation rules to determine the derivative of a function with trigonometric components. As students progress, they will need to understand and combine these techniques to solve increasingly complex problems.