Problem 18 - Thomas Calculus

Question

Find the first derivatives of the functions in Exercises \(11-18\) . $$ k(\theta)=(\sin (\theta+5))^{5 / 4} $$

Step-by-Step Solution

Answer

\( k'(\theta) = \frac{5}{4} (\sin(\theta + 5))^{-3/4} \cdot \cos(\theta + 5) \).

Step 1: Identify the Outer Function

The function given is \( k(\theta)=(\sin (\theta+5))^{5 / 4} \). The outer function here is \( u^{5/4} \), where \( u = \sin(\theta + 5) \). This means we will need to use the chain rule, where the derivative of the outer function \( (u^{5/4}) \) is \( \frac{5}{4} u^{\frac{5}{4} - 1} \).

Step 2: Identify the Inner Function

The inner function in the expression is \( \sin(\theta + 5) \). We will need to take the derivative of this inner function when applying the chain rule.

Step 3: Apply the Chain Rule

According to the chain rule, the derivative of \( k(\theta) = (\sin (\theta+5))^{5 / 4} \) is the derivative of the outer function multiplied by the derivative of the inner function. So, first take the derivative of \( u^{5/4} \):\[ \frac{d}{du} u^{5/4} = \frac{5}{4} u^{1/4}. \]

Step 4: Calculate the Derivative of Inner Function

Now, take the derivative of the inner function \( \sin(\theta + 5) \):\[ \frac{d}{d\theta} \sin(\theta + 5) = \cos(\theta + 5). \]

Step 5: Combine Results

Combine the derivatives from Steps 3 and 4 to compute the full derivative of \( k(\theta) \):\[\frac{dk}{d\theta} = \frac{5}{4} (\sin(\theta + 5))^{1/4} \cdot \cos(\theta + 5).\]Thus, the derivative is:\[k'(\theta) = \frac{5}{4} (\sin(\theta + 5))^{-3/4} \cdot \cos(\theta + 5).\]

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus, used to find the derivative of a composite function. A composite function is essentially a function within another function. To use the chain rule, recognize the outer and inner functions.
The outer function is the one applied last, while the inner function is the first applied. In the exercise, the outer function is \( u^{5/4} \), and the inner function is \( u = \sin(\theta + 5) \). To find the derivative, we differentiate both functions separately.

• First, differentiate the outer function: \( \frac{d}{du} u^{5/4} = \frac{5}{4} u^{1/4} \).
• Next, differentiate the inner function: \( \frac{d}{d\theta} \sin(\theta + 5) = \cos(\theta + 5) \).

Finally, multiply these derivatives to get the final result: \( \frac{dk}{d\theta} \). This multiplication step is crucial in applying the chain rule.

Trigonometric Functions

Trigonometric functions like sine and cosine are critical in calculus. These functions represent relationships in triangles, but they also have properties that make them crucial for analyzing periodic phenomena.
In our exercise, we see \( \sin(\theta + 5) \). Here, \( \sin \) is the trigonometric function, and \( \theta + 5 \) is the angle. When you differentiate a sine function, you get the cosine function. Therefore:

• The derivative of \( \sin(\theta + 5) \) is \( \cos(\theta + 5) \).

Trigonometric identities and derivatives are tools that allow us to break down and understand complex calculus problems.

Calculus Problem Solving

Solving calculus problems involves breaking a complex problem into smaller, more straightforward steps. Here’s a simple approach:

• Start by identifying the types of functions involved. In our case, these are power functions and trigonometric functions.
• Use the chain rule whenever you deal with composite functions.
• Perform each differentiation step separately, before combining them.

It's important to keep the order of operations clear and not rush through differentiations.
This structured approach not only simplifies solving calculus problems but also ensures accuracy. Developing these problem-solving skills is invaluable for tackling a wide range of mathematical challenges.