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When we have a function that is the product of two smaller functions, we use the product rule to differentiate it. The product rule is important because it lets us find derivatives of more complex functions easily. The rule states: \((uv)' = u'v + uv'\). This means that the derivative of two functions multiplied together \(u\) and \(v\) is \(u'v\) plus \(uv'\). Here is how it works:

• \(u\): One of the functions you are multiplying, such as \(\sec x\) in our case.
• \(v\): The other function you are multiplying, which for our problem is \(\tan x\).
• \(u'\): The derivative of \(u\), or \(\sec x\tan x\) in this exercise.
• \(v'\): The derivative of \(v\), or \(\sec^2 x\) here.

This step-by-step approach given by the product rule is a systematic way to handle products of functions while differentiating. It's one of the core tools in calculus, especially when dealing with intertwined functions. Break complex expressions down into products, and apply this golden rule. This allows for easier calculation and understanding.

Trigonometric functions, like \(\sec x\) and \(\tan x\), appear frequently in calculus due to their unique properties and periodic nature. They are the backbone of many calculus problems both simple and complex.

• \(\sec x\): Also known as the secant function, is defined as \(1/\cos x\). It's useful because it naturally complements the cosine function and appears as a derivative in calculus problems.
• \(\tan x\): The tangent function is the ratio of sine to cosine, \(\sin x / \cos x\). This function offers a different perspective on angle measurements and slopes, crucial for calculus.

In the context of derivatives:

• Derivative of \(\sec x\): The derivative is \(\sec x \tan x\). This result comes from the product of secant and tangent, reflecting their inherent relationship.
• Derivative of \(\tan x\): This is \(\sec^2 x\). An important derivative that helps describe how the tangent function changes.

Understanding these trigonometric relationships and derivatives is fundamental. They help simplify complex function differentiation, which is essential in many fields, such as physics and engineering.

Differentiation is about finding how a function changes at any point. The approach usually involves systematic steps that you can follow to simplify the process.

Step-by-step Differentiation

Using the product rule along with the knowledge of derivatives, you can easily tackle derivative problems involving trigonometric functions. Here's a breakdown:

• Identify the Functions: Recognize that your function \(f(x) = \sec x \tan x\) is the product of \(u = \sec x\) and \(v = \tan x\).
• Find Derivatives of Each Part: Compute \(u' = \sec x \tan x\) and \(v' = \sec^2 x\).
• Apply the Product Rule: Substitute these into the product rule to get \(f'(x) = (\sec x \tan x)(\tan x) + (\sec x)(\sec^2 x)\).
• Simplify: Combine and simplify the terms to get \(f'(x) = \sec x \tan^2 x + \sec^3 x\).

These steps highlight the clarity of following a structured method for differentiation. This makes solving even complex problems approachable and manageable. With practice, these steps become intuitive, allowing you to swiftly identify the necessary actions for any given function.