Problem 3 - Calculus

Question

Find \(D_{x} y .\) \(y=\sin ^{2} x+\cos ^{2} x\)

Step-by-Step Solution

Answer

The derivative is 0.

Step 1: Understand the Problem

We're asked to find the derivative with respect to x of the function \(y = \sin^2 x + \cos^2 x\). This involves using derivative rules and identities.

Step 2: Identify Trigonometric Identity

We use the trigonometric identity: \(\sin^2 x + \cos^2 x = 1\). Recognizing this simplifies our problem, as it allows us to rewrite \(y\) in a simpler form.

Step 3: Simplify the Expression

Since \(\sin^2 x + \cos^2 x = 1\), we can substitute this into our function, giving us \(y = 1\).

Step 4: Differentiate the Constant

The derivative of a constant is zero. Thus, the derivative of \(y = 1\) with respect to \(x\) is \(D_{x} y = 0\).

Key Concepts

Understanding Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are always true for every value of the occurring variables. They are instrumental in simplifying complex trigonometric expressions.

• One of the most fundamental identities is the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \]This identity tells us that the square of the sine of any angle added to the square of the cosine of the same angle always equals one.
• Knowing these identities allows mathematicians to simplify expressions and makes solving calculus problems easier, as seen in our exercise.

Using the identity, we can see how a seemingly complex trigonometric expression simplifies beautifully to a constant. Recognizing these identities contributes to a deeper understanding of trigonometric functions and their relationships.

The Basics of Derivatives

Derivatives are a fundamental concept in calculus, representing the rate of change of a function with respect to a variable. In simpler terms, the derivative shows how a function changes as its input changes. This concept is crucial for understanding dynamic systems and changes over time.

• The derivative of a constant is always zero. This is because constants do not change, thus they have no rate of change.
• The power rule is often used: for a function \(f(x) = x^n\), the derivative is \(f'(x) = nx^{n-1}\).

In our scenario, since the expression simplifies to a constant \( y = 1 \), the derivative with respect to \(x\) is zero. Recognizing when an expression simplifies to a constant can save time and effort in calculus problems.

Simplification Techniques in Calculus

Simplification techniques are strategies used to make mathematical expressions easier to work with. In calculus, simplifying expressions can help solve problems more efficiently and with less complexity.

• Identifying and using known identities, like the trigonometric identities we've discussed, is a powerful simplification strategy.
• Simplifying complex expressions can help identify constants, which play a critical role when finding derivatives, as they often simplify the differentiation process.

In the given exercise, recognizing the trigonometric identity allowed for simplification from \( \sin^2 x + \cos^2 x \) to 1, leading us to an easy differentiation step. Such techniques are crucial not just for solving calculus problems more efficiently, but also for deepening understanding of the relationships between different mathematical concepts.