Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\sin x \tan x $$

Step-by-Step Solution

Answer

The derivative is: \( D_x y = \sin x + \sin x \sec^2 x \).

Step 1: Identify the Form Needed for Differentiation

To differentiate \( y = \sin x \tan x \) with respect to \( x \), we need to recognize that this is a product of two functions: \( u = \sin x \) and \( v = \tan x \). Therefore, we will use the product rule for differentiation.

Step 2: State the Product Rule

The product rule states that if \( y = u \cdot v \), then the derivative \( D_x y \) is given by: \[ D_x y = u' v + u v' \]. This means we need to find the derivatives of \( u \) and \( v \) separately and then apply this rule.

Step 3: Differentiate Each Function

Differentiate \( u = \sin x \) with respect to \( x \): \[ u' = \cos x \].Differentiate \( v = \tan x \) with respect to \( x \): \[ v' = \sec^2 x \].

Step 4: Apply the Product Rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the product rule formula: \[ D_x y = (\cos x)(\tan x) + (\sin x)(\sec^2 x) \].

Step 5: Simplify the Expression

This expression can be simplified using trigonometric identities. Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \sec x = \frac{1}{\cos x} \): \[ D_x y = (\cos x)\left(\frac{\sin x}{\cos x}\right) + (\sin x)\left(\frac{1}{\cos^2 x}\right) \].After simplification, we get: \[ D_x y = \sin x + \frac{\sin x}{\cos^2 x} \].Further simplify by combining terms: \[ D_x y = \sin x + \sin x \sec^2 x \].

Key Concepts

Product Rule

The product rule is a fundamental theorem in calculus used for finding the derivative of the product of two functions. When you have two functions, say \( u(x) \) and \( v(x) \), and you want to differentiate their product \( y = u(x)v(x) \), the product rule offers a straightforward approach. It tells us that the derivative of this product \( y \) with respect to \( x \) is given by:

• \( D_x y = u'(x) v(x) + u(x) v'(x) \)

This means that instead of taking the derivative of the entire product directly, you can differentiate each function separately and combine them to get the derivative of the product.

For example, if \( u(x) = \sin x \) and \( v(x) = \tan x \), as in our exercise, we use the product rule to find the derivative of their product \( \sin x \tan x \). This method simplifies the process and ensures accuracy in differentiation.

Trigonometric Functions

Trigonometric functions are a set of functions in mathematics that relate angles in a right triangle to ratios of two side lengths. These functions include sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), among others, and they are fundamental in many areas of mathematics, especially calculus.

In the given exercise, we deal with two specific trigonometric functions: the sine function and the tangent function. Understanding these functions and their properties is crucial:

• The sine function \( \sin x \) relates the angle to the ratio of the opposite side and hypotenuse in a right triangle.
• The tangent function \( \tan x \) is the ratio of the sine and cosine functions, expressed as \( \tan x = \frac{\sin x}{\cos x} \).

These functions are periodic, meaning they repeat their values in regular intervals, which makes them useful in modeling oscillatory phenomena like waves.

Derivative of Trigonometric Functions

The derivative of a trigonometric function is a way to determine how the function changes with respect to its variable, typically an angle \( x \). This concept is crucial when we perform calculations involving the rates of change, such as in physics or engineering.

To differentiate trigonometric functions like \( \sin x \) and \( \tan x \), we use standard differentiation techniques:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \tan x \) is \( \sec^2 x \), where \( \sec x = \frac{1}{\cos x} \).

Recognizing these derivatives is important as it allows you to use the product rule effectively when differentiating combinations of trigonometric functions.

By applying these derivative rules to the functions in our exercise, you enable the application of the product rule formula to find the overall derivative efficiently.