Problem 37 - Calculus Early Transcendentals

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, as the derivative \(f'(x) = \sec^2 x\).

Step 1: Understanding the given statement

The problem statement presents the equation \(f(x) \cos x = \sin x\). It asks whether the derivative of \(f(x)\) is \(\sec^2(x)\). Let us first analyze the given function equation.

Step 2: Isolate the function \(f(x)\)

We rearrange the equation \(f(x) \cos x = \sin x\) to express \(f(x)\) in terms of \(x\). Divide both sides by \(\cos x\), assuming \(\cos x eq 0\):\[ f(x) = \frac{\sin x}{\cos x} = \tan x \]

Step 3: Differentiate \(f(x) = \tan x\)

Now that we have \(f(x) = \tan x\), find the derivative \(f'(x)\) using the rule \(\frac{d}{dx}(\tan x) = \sec^2 x\):\[ f'(x) = \sec^2 x \]

Step 4: Verify the statement

Since we derived that \(f'(x) = \sec^2 x\) from \(f(x) = \tan x\), and this matches the statement's claim, the original statement is true.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus representing the rate of change of a function with respect to a variable. It's essentially about finding how a function changes as its input changes. The derivative gives us the slope of the function at any point along its graph.

• In the given problem, we began with a function equation involving trigonometric identities: \(f(x) \cos x = \sin x\).
• From this, we isolated \(f(x)\) and identified it as \(\tan x\), a function whose rate of change could be determined.
• The derivative, \(f'(x)\), which is \(\sec^2 x\), tells us how \(\tan x\) behaves as \(x\) varies.

Grasping the concept of derivatives helps in understanding various phenomena such as velocity, acceleration, and other rates of change, transforming both mathematical and real-world problems into manageable forms.

Trigonometric Functions

Trigonometric functions are foundational in calculus and mathematics. They describe relationships between the angles and sides of triangles and extend to modeling periodic phenomena.
Some major trigonometric functions include:

• Sine \(\sin x\)
• Cosine \(\cos x\)
• Tangent \(\tan x\)

With the specific function \(\tan x = \frac{\sin x}{\cos x}\) in this problem, trigonometric identities are utilized to simplify and solve the equation. These functions reoccur across different branches of math due to their cyclical patterns and are applied in various fields, from engineering to physics.
Understanding them is crucial for tackling more complex calculus problems.

Function Differentiation

Differentiation is the process of finding a derivative. It allows us to determine the rate at which a function value changes concerning its input value.

• This process involves applying specific differentiation rules based on the function type, such as power, product, chain, and quotient rules.
• For \(f(x) = \tan x\), given its identity, we applied the differentiation rule for tangent: \(\frac{d}{dx}(\tan x) = \sec^2 x\).
• The statement verified the same derivative outcome, aiding in authenticating original claims.

The ability to differentiate functions accurately is vital for solving diverse problems in calculus, ensuring a functional understanding of changes and dynamics inherent in mathematical models.