Problem 17 - Calculus Single Variable

Question

Differentiate the given expression with respect to \(x\). \(\tan (x) \sec (x)\)

Step-by-Step Solution

Answer

The derivative is \( \sec^3(x) + \sec(x) \tan^2(x) \).

Step 1: Use the Product Rule

The expression is a product of two functions: \( u(x) = \tan(x) \) and \( v(x) = \sec(x) \). To differentiate \( u(x)v(x) \), we use the product rule: \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).

Step 2: Differentiate \(u(x) = \tan(x)\)

The derivative of \(u(x) = \tan(x)\) is \(u'(x) = \sec^2(x)\).

Step 3: Differentiate \(v(x) = \sec(x)\)

The derivative of \(v(x) = \sec(x)\) is \(v'(x) = \sec(x)\tan(x)\).

Step 4: Apply the Product Rule

Plug the derivatives and original functions into the product rule formula: \( \frac{d}{dx}[\tan(x) \sec(x)] = \sec^2(x) \cdot \sec(x) + \tan(x) \cdot \sec(x) \cdot \tan(x) \).

Step 5: Simplify the Expression

The expression simplifies to: \( \sec^3(x) + \sec(x) \tan^2(x) \).

Key Concepts

Product Rule

When faced with differentiating the product of two functions, we use the product rule. The product rule is a fundamental tool in calculus differentiation. It is used when you have two functions being multiplied together, and you need to find the derivative of their product. Here’s how you apply it:

• Identify the two functions being multiplied. Let’s call them \( u(x) \) and \( v(x) \).
• The formula for the product rule is: \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).
• This essentially means you take the derivative of the first function and multiply it by the second function, then add the product of the first function and the derivative of the second function.

This rule makes it easier to handle more complex functions by breaking them down into simpler parts. Always remember to identify each function correctly and follow the formula step by step to avoid confusion.

Derivative of Trigonometric Functions

Trigonometric functions like \( \tan(x) \) and \( \sec(x) \) have special derivatives that are widely used in calculus. Knowing these derivatives is crucial for solving problems where these functions appear:

• The derivative of \( \tan(x) \) is \( \sec^2(x) \). This means when you differentiate \( \tan(x) \), you get \( \sec(x) \cdot \sec(x) \).
• The derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \). This derivative shows that you have to multiply \( \sec(x) \) by \( \tan(x) \) after taking the derivative.

Learning these by heart makes it faster and easier to work through differentiation problems involving trigonometric functions. If you remember these derivatives, they will naturally apply when you work through trigonometric differentiation tasks.

Differentiation Techniques

Differentiation techniques in calculus are varied methods used to find the derivative of a function. There are basic rules like the power rule, and more advanced ones like chain rule, product rule, and quotient rule:

• The power rule is typically used for functions where a variable is raised to a constant power.
• The product rule, as explained earlier, is used when differentiating the product of two functions.
• For complex functions, the chain rule helps when dealing with compositions, and the quotient rule is vital for the ratios of two functions.

Each technique has its place depending on the structure of the function you are differentiating. In practice, knowing which rule to apply and when can greatly simplify the solution and ensure accuracy. It's also helpful to sometimes combine these techniques for more complicated problems. With practice, choosing the right technique becomes second nature, allowing you to tackle a wide variety of calculus problems confidently.