Problem 40 - Calculus Single Variable

Question

Use the Product Rule to differentiate the given expression with respect to \(x\). \(\tan (x)+\tan ^{2}(x)\)

Step-by-Step Solution

Answer

The derivative is \( \sec^2(x) (1 + 2\tan(x)) \).

Step 1: Identify the Functions

We have the expression \(\tan(x) + \tan^2(x)\), which we need to differentiate with respect to \(x\). This can be viewed as a sum of two functions: \(f(x) = \tan(x)\) and \(g(x) = \tan^2(x)\).

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

The derivative of \( \tan(x) \) with respect to \( x \) is \( \sec^2(x) \). So, \( f'(x) = \sec^2(x) \).

Step 3: Differentiate \( g(x) = \tan^2(x) \) Using the Chain Rule

To differentiate \( \tan^2(x) \), set \( u = \tan(x) \) so \( g(x) = u^2 \). Then by the chain rule, \( g'(x) = 2u \cdot \frac{du}{dx} = 2\tan(x) \cdot \sec^2(x) \).

Step 4: Add the Derivatives Together

Since differentiation is linear, we can differentiate term by term. The derivative of the entire expression is the sum of the individual derivatives: \( \sec^2(x) + 2\tan(x) \sec^2(x) \).

Step 5: Factor Out Common Terms (Optional)

Notice that both terms of the derivative contain \( \sec^2(x) \). Factoring \( \sec^2(x) \) gives: \( \sec^2(x)(1 + 2\tan(x)) \).

Key Concepts

Differentiation Basics

Differentiation is a fundamental concept in calculus. It allows us to find the derivative of a function, which represents the rate of change. When we differentiate with respect to a variable, like \( x \), we are essentially looking at how a function changes as \( x \) changes.

In our exercise, we need to differentiate the expression \( \tan(x) + \tan^2(x) \). The derivative gives us insight into the behavior of the function and its slope at any given point. This is a crucial skill in calculus, as it helps in understanding the properties and motions of functions. To differentiate a sum like \( \tan(x) + \tan^2(x) \), we can differentiate each part separately and then add the results together. This principle leverages the linearity of differentiation, making our task more manageable.

Applying the Chain Rule

The Chain Rule is an essential tool for differentiating composite functions. If you encounter a function that is a composition of two or more functions, the Chain Rule is your go-to strategy. The rule states that to differentiate a composite function \( f(g(x)) \), you multiply the derivative of the outside function \( f \) at \( g(x) \) by the derivative of the inside function \( g(x) \).

In our exercise, when differentiating \( \tan^2(x) \), we apply the Chain Rule. Here, we set \( u = \tan(x) \), which makes our function a composite: \( u^2 \). By the Chain Rule, the derivative is \( 2u \cdot \frac{du}{dx} \). Since \( \frac{du}{dx} = \sec^2(x) \) for \( u = \tan(x) \), the derivative becomes \( 2\tan(x) \cdot \sec^2(x) \).

This application shows how the Chain Rule simplifies differentiation in cases involving nested functions, ensuring accuracy in calculating the derivative.

Understanding Trigonometric Functions in Calculus

Trigonometric functions like \( \tan(x) \), \( \sin(x) \), and \( \cos(x) \) often appear in calculus problems due to their periodic nature and fundamental properties. Understanding how to differentiate these functions is crucial.

For instance, the derivative of \( \tan(x) \) is \( \sec^2(x) \). This is derived from the quotient of the sine and cosine functions since \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Differentiating each basic trig function provides indispensable tools for tackling more complex calculus problems.

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) involves \( \sec^2(x) \), showcasing how derivatives of basic trig functions can yield more complex trigonometric identities.

Such transformations and derivatives are vital in physics, engineering, and any field requiring precise motion analysis or wave behavior understanding. Recognizing and manipulating these trigonometric derivatives allows for deeper insights into function behaviors across diverse applications.