Problem 10 - University Calculus: Early Transcendentals

Question

Find \(d y / d x\). $$y=(\sin x+\cos x) \sec x$$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \tan x + \cos x - \sin x\).

Step 1: Expand the Function

Start by rewriting the given function \[y=(\sin x + \cos x) \sec x\]as a product of two functions: \(u = \sin x + \cos x\) and \(v = \sec x\). Then, express the multiplication explicitly as:\[y = u \cdot v\].

Step 2: Apply the Product Rule

To find \(\frac{dy}{dx}\), use the product rule of differentiation, which states:\[\frac{d}{dx}(u \, v) = u \frac{dv}{dx} + v \frac{du}{dx}\].In this case, identify \(u = \sin x + \cos x\) and \(v = \sec x\).

Step 3: Differentiate Each Component

First, differentiate \(u = \sin x + \cos x\) to get:\[\frac{du}{dx} = \cos x - \sin x\].Then differentiate \(v = \sec x\) to get:\[\frac{dv}{dx} = \sec x \tan x\].

Step 4: Calculate the Derivative Using Product Rule

Substitute the derivatives from Step 3 into the product rule formula:\[\frac{dy}{dx} = (\sin x + \cos x)(\sec x \tan x) + (\sec x)(\cos x - \sin x)\].

Step 5: Simplify the Expression

Simplify the expression from Step 4:- For \((\sin x + \cos x)(\sec x \tan x)\), break it down: \[\sin x \sec x \tan x + \cos x \sec x \tan x = \tan x + \tan x\cos^2 x = \tan x\].- For \((\sec x)(\cos x - \sin x)\), simplify: \[\cos x - \sin x\].Combine: \[\frac{dy}{dx} = \tan x + (\cos x - \sin x)\].The derivative is therefore:\[\tan x + \cos x - \sin x\].

Key Concepts

Product Rule

In calculus, the product rule is a fundamental method used to differentiate expressions where two functions are multiplied together. This rule is essential when dealing with more complex functions that cannot be simplified.
To understand how it works, consider two functions, denoted as \(u(x)\) and \(v(x)\). The product rule allows us to find the derivative of their product, \( y = u(x) \cdot v(x) \). The formula is given by:

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

Here's the breakdown of the formula:
- Differentiate \(v(x)\) while keeping \(u(x)\) unchanged, and multiply the result by \(u(x)\).- Differentiate \(u(x)\) while keeping \(v(x)\) unchanged, and multiply the result by \(v(x)\).
The sum of these two products gives the derivative of the product of the original functions. This is extremely useful in the original exercise where the expression \((\sin x + \cos x) \sec x\) is given. By carefully applying the product rule, we find the derivative, simplifying our task.

Trigonometric Functions

Trigonometric functions are a core part of calculus and are used in various mathematical problems, including differentiation. Common trigonometric functions include sine \(\sin\), cosine \(\cos\), and tangent \(\tan\).
These functions exhibit periodic behavior and relate angles to the ratios of sides in right-angled triangles.

• \(\sin x\) represents the ratio of the opposite side over the hypotenuse.
• \(\cos x\) represents the ratio of the adjacent side over the hypotenuse.
• \(\tan x\) is the ratio of the opposite side over the adjacent side, or equivalently, \(\tan x = \frac{\sin x}{\cos x}\).

These are crucial when dealing with derivatives of trigonometric expressions, as their derivatives have straightforward patterns:

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

Understanding these relationships aids in solving the exercise, where we leverage the knowledge of these derivatives to apply them in the product rule for differentiation.

Derivatives

Derivatives represent the rate at which a function changes with respect to one of its variables. This rate of change is crucial in understanding how functions behave and is a central concept in calculus.
In simple terms, derivatives tell us how a function slopes or curves at any point. In the exercise given, we are tasked with finding \(\frac{dy}{dx}\), which represents how \(y\) changes as \(x\) changes.
Calculating derivatives can involve several rules and methods, including:

• The Power Rule: To differentiate \(x^n\), use \(nx^{n-1}\).
• The Chain Rule: For composite functions, \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).
• The Product Rule: As used in the original exercise.
• The Quotient Rule: Useful when functions are divided.

In calculating the derivative of \(y = (\sin x + \cos x) \sec x\), we used the product rule. Differentiating each part, combining the results, and simplifying, allows us to find the precise rate of change for the given function expression. This highlights the importance of derivatives in analyzing and predicting behaviors of mathematical models.