Problem 7 - University Calculus: Early Transcendentals

Question

Find \(d y / d x\). $$f(x)=\sin x \tan x$$

Step-by-Step Solution

Answer

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

Step 1: Identify and Label Functions

Given the function \( f(x) = \sin x \tan x \), identify that it is a product of two functions: \( u = \sin x \) and \( v = \tan x \).

Step 2: Recall the Product Rule

The product rule for differentiation states that if \( f(x) = u(x)v(x) \), then \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \). We will apply this rule to differentiate \( f(x) = \sin x \tan x \).

Step 3: Differentiate \( u(x) = \sin x \)

The derivative of \( \sin x \) is \( \cos x \). Thus, \( u'(x) = \cos x \).

Step 4: Differentiate \( v(x) = \tan x \)

The derivative of \( \tan x \) is \( \sec^2 x \). Thus, \( v'(x) = \sec^2 x \).

Step 5: Apply the Product Rule

Using the product rule: \( \frac{d}{dx}[\sin x \tan x] = \cos x \cdot \tan x + \sin x \cdot \sec^2 x \).

Step 6: Simplify the Result

Substitute the derivatives into the product rule formula: \( \frac{d}{dx}[\sin x \tan x] = \cos x \tan x + \sin x \sec^2 x \). There may not be further simplification that makes the expression simpler or cleaner.

Key Concepts

Product Rule

The product rule is a fundamental concept in calculus differentiation, making it possible to find the derivative of a product of two functions. If you have two functions, say \( u(x) \) and \( v(x) \), the product rule states that the derivative of their product \( u(x)\cdot v(x) \) can be calculated as:

• \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \)

This formula relies on computing the derivatives of each function separately and then combining them in a specific way. It ensures we account for how both functions change, which allows us to properly understand their combined rate of change. This rule is invaluable when tackling more complex differentiation problems, particularly those involving trigonometric functions or functions defined as products.

Trigonometric Derivatives

Trigonometric functions like sine, cosine, tangent, and others have specific rules that help us differentiate them. Their derivatives are essential building blocks in calculus. Understanding these rules is crucial for solving differentiation problems involving trigonometric expressions. Here are the basic derivatives you will encounter frequently:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).
• The derivative of \( \tan x \) is \( \sec^2 x \).

These rules allow us to find how fast these trigonometric functions are changing with respect to the variable \( x \). Once you are familiar with these derivatives, applying them within the context of the product rule becomes much easier.

Derivative of Sine

The sine function, \( \sin x \), is one of the simplest trigonometric functions to differentiate. Its derivative is straightforward: the rate of change of \( \sin x \) is \( \cos x \). This can be derived from the limit definition of the derivative and a fundamental understanding of the unit circle in trigonometry.
This derivative is especially handy in ongoing calculations because it transforms the sine function into a cosine function, which is directly related via phase shift. The simplicity of this derivative makes it a common feature in many calculus problems, including those involving the product rule.

Derivative of Tangent

The derivative of the tangent function is a bit more complex than that of sine. The tangent function, \( \tan x \), is defined as \( \frac{\sin x}{\cos x} \). To find its derivative, an understanding of the quotient rule, another powerful differentiation tool, is required. However, the derivative of \( \tan x \) simplifies to \( \sec^2 x \), leveraging the identity \( \sec x = \frac{1}{\cos x} \).
This derivative reveals how rapidly the tangent function's slope changes as its input changes, and it is particularly useful when dealing with angular motion and oscillations where rapid changes are involved. Knowing the derivative \( \sec^2 x \) means we can quickly apply this in various calculus contexts, notably, when solving problems using the product rule.