Problem 7 - Calculus - AP Edition

Question

Find \(f^{\prime}(x)\). $$f(x)=\sec x-\sqrt{2} \tan x$$

Step-by-Step Solution

Answer

\(f^{\prime}(x) = \sec x \tan x - \sqrt{2} \sec^2 x\).

Step 1: Identify the Derivatives Needed

We need the derivatives of \(\sec x\) and \(\tan x\). The derivative of \(\sec x\) is \(\sec x \tan x\), and the derivative of \(\tan x\) is \(\sec^2 x\).

Step 2: Apply the Derivative to Each Term

The function \(f(x) = \sec x - \sqrt{2} \tan x\) consists of two terms. We'll find the derivative of each separately. The derivative of \(\sec x\) is \(\sec x \tan x\), and the derivative of \(-\sqrt{2} \tan x\) is \(-\sqrt{2} \sec^2 x\).

Step 3: Combine the Derivatives

Add the derivatives from each term. This gives us the derivative of the entire function:\\(f^{\prime}(x) = \sec x \tan x - \sqrt{2} \sec^2 x\).

Key Concepts

Product Rule

When you have a function that is the product of two other functions, you'll need to use the product rule to find its derivative. This rule is a fundamental tool in calculus differentiation. The product rule states that if you have two functions, say \( u(x) \) and \( v(x) \), then the derivative of their product \( u(x) \, v(x) \) is given by:\[(uv)' = u'v + uv'\]This means you take the derivative of the first function, multiply it by the second function, and then add the product of the first function and the derivative of the second function.

• Identify the two functions involved.
• Differentiate each function separately.
• Follow the formula to compute the derivative of the product.

Understanding the product rule is crucial, especially when dealing with more complex functions in calculus, such as when learning the chain rule or differentiating trigonometric functions.

Chain Rule

The chain rule is a powerful technique for differentiating composite functions. This is when a function is "inside" another function, such as \( f(g(x)) \). The chain rule provides a way to differentiate such functions smoothly.The formula for the chain rule is given by:\[(f(g(x)))' = f'(g(x)) \, g'(x)\]Here's a step-by-step approach on how to use the chain rule:

• First, differentiate the outer function, keeping the inner function unchanged.
• Then, multiply this derivative by the derivative of the inner function.

Applying the chain rule allows you to tackle complex derivatives that involve multiple layers of functions, making it possible to differentiate functions more easily than one might expect at first glance.

Calculus Differentiation

Differentiation is one of the core operations in calculus, focusing on finding the rate at which a function is changing at any given point. This involves calculating the derivative, which is a function that gives the slope of the original function's tangent line at any point.When differentiating, whether you're dealing with basic polynomial functions or more intricate trigonometric ones like \( \sec x \) and \( \tan x \), as in the original problem, the key is to apply the right rules:

• The product rule for products of functions.
• The chain rule for composite functions.
• Basic differentiation rules such as power, sum, or difference rules for simpler functions.

Calculus differentiation provides valuable insights into the behaviour of functions, helping to determine maxima, minima, and inflection points, among other critical aspects of a function's graph. Understanding the concepts of differentiation opens doors to solving a wide array of mathematical problems.