Problem 175 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d y}{d x}\) for the given functions. $$y=x^{2}-\sec x+1$$

Step-by-Step Solution

Answer

\( \frac{d y}{d x} = 2x - \sec x \tan x \).

Step 1: Differentiate Each Term Separately

To find \( \frac{d y}{d x} \), differentiate each term of the function \( y = x^2 - \sec x + 1 \) separately with respect to \( x \).

Step 2: Differentiate \(x^2\)

The derivative of \( x^2 \) with respect to \( x \) is \( 2x \).

Step 3: Differentiate \(-\sec x\)

The derivative of \(-\sec x\) with respect to \( x \) is \(-\sec x \tan x\).

Step 4: Differentiate Constant Term

The derivative of a constant, such as \( 1 \), is \( 0 \).

Step 5: Combine the Derivatives

Combine the derivatives of each term to find the derivative of \( y \). Therefore, \( \frac{d y}{d x} = 2x - (\sec x \tan x) + 0 = 2x - \sec x \tan x \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It allows us to understand how functions change. Think of it as a way to find the rate at which a function's value changes with respect to change in another variable. In simple terms, differentiation helps us know how steep or flat a curve is at any given point. The process of differentiation gives us what is known as a "derivative." When you differentiate a function, you are finding its derivative. In mathematical notation, the derivative of a function is often represented as \( \frac{dy}{dx} \), where \( y \) is the function and \( x \) is the variable. Here are some important points about differentiation:

• The derivative of a constant is always zero, because constants do not change.
• The derivative of \( x^n \), where \( n \) is a constant, is \( nx^{n-1} \).
• You can differentiate each term of a function separately when the function is a sum or difference of terms.

Trigonometric Functions

Trigonometric functions play a significant role in differentiation, especially in calculus problems involving angles or periodic phenomena. Common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), secant (\( \sec \)), and tangent (\( \tan \)). These functions have specific behaviors and derivatives that are important to remember:

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

Trigonometric functions often appear in problems involving angles or oscillations. Understanding their derivatives helps in solving many practical and theoretical problems in physics, engineering, and other fields.

Chain Rule

The chain rule is an essential technique in differentiation. It is used when finding the derivative of composite functions, where one function is nested inside another. In more straightforward terms, if you have a function inside another function, the chain rule helps you differentiate it. The chain rule formula can be expressed as:\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]where \( y \) is a function of \( u \), and \( u \) is a function of \( x \). The chain rule breaks down the differentiation into manageable steps:

• First, differentiate the outer function with respect to the inner function.
• Then, multiply this result by the derivative of the inner function with respect to \( x \).

It's like peeling an onion layer by layer, differentiating the outer layer first before moving inward. This approach is very useful in finding derivatives of complex functions, ensuring precise and accurate results.