Problem 343 - Calculus Volume 1

Question

For the following exercises, find \(f^{\prime}(x)\) for each function. $$f(x)=\log (\sec x)$$

Step-by-Step Solution

Answer

The derivative \(f'(x) = \tan x\).

Step 1: Identify the Function

The given function is \(f(x) = \log(\sec x)\). This means that we have a logarithmic function involving \(\sec x\). To find the derivative, we'll need to use the chain rule since the argument of the logarithm is more complex than just \(x\).

Step 2: Recall Derivative Rules

Remember the derivative rules: the derivative of \(\log(u)\) is \(\frac{1}{u} \cdot u'\), and the derivative of \(\sec x\) is \(\sec x \cdot \tan x\). We will apply these rules as we differentiate step-by-step.

Step 3: Apply Chain Rule for Differentiation

Using the chain rule, the derivative of \(f(x) = \log(\sec x)\) is \(f'(x) = \frac{1}{\sec x} \cdot (\sec x \cdot \tan x)\). This step arises because the chain rule states that the derivative of \(\log(u)\) is \(\frac{1}{u}\) times the derivative of \(u\), where \(u = \sec x\) in this case.

Step 4: Simplify the Expression

After differentiating, we have \(f'(x) = \frac{1}{\sec x} \cdot (\sec x \cdot \tan x)\). Simplify by cancelling \(\sec x\) in the numerator and denominator, resulting in \(f'(x) = \tan x\).

Key Concepts

chain rule

The chain rule is a fundamental technique in calculus that allows us to differentiate composite functions. When you have a function composed of other functions, like \( f(x) = \log(\sec x) \), you need to use the chain rule to find the derivative. In simple terms, the chain rule breaks down the process of differentiation into separate parts that can be managed individually.

Here's how it works: If you have a function \( y = g(f(x)) \), the chain rule states that the derivative \( y' \) is the derivative of \( g \) with respect to \( f(x) \) multiplied by the derivative of \( f(x) \) with respect to \( x \). In formula form, this is expressed as \( y' = g'(f(x)) \cdot f'(x) \).

Consider our function \( f(x) = \log(\sec x) \):

• The outer function is \( \log(u) \),
• The inner function is \( u = \sec x \).

To differentiate \( \log(\sec x) \), we first differentiate the outer function \( \log(u) \) with respect to \( u \), which gives \( \frac{1}{u} \). Then we differentiate the inner function \( \sec x \) with respect to \( x \), resulting in \( \sec x \cdot \tan x \). Finally, multiply these components together: \( \frac{1}{\sec x} \cdot (\sec x \cdot \tan x) \), which simplifies to \( \tan x \).

trigonometric derivatives

Trigonometric derivatives play a crucial role in calculus, especially when dealing with functions that involve trigonometric terms like sine, cosine, and secant. Understanding these derivatives is essential for solving many problems in calculus.

Let's focus on the derivative rules for some key trigonometric functions:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).
• The derivative of \( \sec x \) is \( \sec x \cdot \tan x \).

In our exercise, we differentiated a function involving the secant. Knowing that the derivative of \( \sec x \) is \( \sec x \cdot \tan x \) allows us to apply it directly when we find the derivative of logarithmic functions like \( \log(\sec x) \).

These derivatives hold the key to breaking down complex expressions into more manageable parts, illustrating the importance of understanding the derivatives of basic trigonometric functions for calculus exercises.

calculus exercises

Calculus exercises often involve applying various rules and techniques to find derivatives, integrals, and limits. These exercises strengthen your understanding of the fundamental principles of calculus, helping build a solid foundation for more advanced mathematics and real-world applications.

When tackling calculus problems, it's crucial to:

• Identify the type of function you are dealing with.
• Recognize which rules apply for differentiation or integration.
• Work through the steps systematically, simplifying as you go.

In our given exercise, the goal was to differentiate \( f(x) = \log(\sec x) \). We started by identifying the function as a composition of a logarithmic and trigonometric function, thus invoking the chain rule.

Calculus exercises challenge you to combine different rules, such as the chain rule and trigonometric derivatives, facilitating deeper understanding and proficiency in solving complex mathematics problems. This approach is not only crucial for academic success but also for practical problem-solving in various scientific and engineering fields.