Problem 37 - Precalculus

Question

Determine whether the statement is true or false. Explain your answer. If \(f(x) \cos x=\sin x,\) then \(f^{\prime}(x)=\sec ^{2} x\)

Step-by-Step Solution

Answer

The statement is true; both derivatives match.

Step 1: Understand the Given Relation

We are given the equation \( f(x) \cos x = \sin x \). The task is to find the derivative \( f'(x) \) based on this equation.

Step 2: Isolate f(x)

Rewrite the equation as \( f(x) = \frac{\sin x}{\cos x} \). Simplifying this gives \( f(x) = \tan x \) since \( \frac{\sin x}{\cos x} \) is the definition of \( \tan x \).

Step 3: Differentiate f(x)

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

Step 4: Evaluate the Given and Derived Derivative

We are given that \( f'(x) = \sec^2 x \). Through differentiation, we've calculated \( f'(x) = \sec^2 x \). Both match, implying the original statement is true.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially in calculus, for describing relationships involving angles and sides in triangles. In trigonometry, common functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions relate angles of a right triangle to the ratios of its sides.

• \( \sin x \) expresses the ratio of the opposite side to the hypotenuse in a right triangle.
• \( \cos x \) represents the ratio of the adjacent side to the hypotenuse.
• \( \tan x \) is the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \).

Understanding these relationships is crucial for solving problems involving angles, waves, and oscillations in various fields. In our exercise, simplifying \( f(x) \cos x = \sin x \) led us to \( f(x) = \tan x \), highlighting the importance of trigonometric identities.

Derivatives

A derivative represents how a function changes as its input changes. It's a fundamental concept in calculus and is used to find slopes of curves or rates of change. Calculating derivatives involves applying rules to differentiate a function.

• For basic functions, like powers of \( x \), rules such as the power rule \( \frac{d}{dx}x^n = nx^{n-1} \) are used.
• For trigonometric functions, specific derivatives apply, such as \( \frac{d}{dx} \sin x = \cos x \) and \( \frac{d}{dx} \tan x = \sec^2 x \).

In the exercise, we found \( f'(x) \) by differentiating \( f(x) = \tan x \). The derivative was \( \sec^2 x \), confirming the original statement. This derivative tells us the rate at which \( \tan x \) changes with respect to \( x \).

Calculator Differentiation

Calculator differentiation involves using tools like graphing calculators or software to compute derivatives efficiently. This is useful, especially for complex functions or when checking manual calculations.

• These tools can handle a wide range of functions, providing instantaneous results.
• They are invaluable for verifying manual work, ensuring accuracy in derivative calculations.
• For educational purposes, they can also graph derivatives, helping visualize how functions change.

In this context, calculators can be used to confirm that \( \frac{d}{dx} \tan x = \sec^2 x \), providing a visual or numerical confirmation of the differentiation result. While understanding the manual process is key, tools like these aid in learning and precision.