Problem 15 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\sin ^{2} x+\cos ^{2} x $$

Step-by-Step Solution

Answer

\(f'(x) = 0\) because \(\sin^{2}x + \cos^{2}x = 1\).

Step 1: Understanding the Problem

We need to find the derivative of the function \(f(x) = \sin^{2}x + \cos^{2}x\). The goal is to compute \(f'(x)\), the derivative of \(f(x)\) with respect to \(x\).

Step 2: Recall Pythagorean Identity

The function \(f(x) = \sin^{2}x + \cos^{2}x\) can be simplified using the Pythagorean identity. The identity states that \(\sin^{2}x + \cos^{2}x = 1\) for any angle \(x\). Thus, \(f(x) = 1\).

Step 3: Differentiate the Simplified Function

Now, since \(f(x) = 1\), we need to find the derivative of a constant function. We know that the derivative of any constant is 0. Therefore, \(f'(x) = 0\).

Key Concepts

Understanding Derivatives

In calculus, derivatives are a fundamental concept that helps us understand how functions change. When we talk about the derivative of a function, denoted as \( f'(x) \), we are essentially looking at the rate at which the function's value changes as the input \( x \) changes. It's like asking, "If I tweak \( x \) a bit, how does \( f(x) \) respond?"
One way to think of the derivative is as the slope of the tangent line to the curve of the function at any point. This slope tells us whether the function is increasing or decreasing at that particular point, and by how much.
To find derivatives, we often use rules like the power rule, product rule, and chain rule, among others, which guide us in breaking down complex functions into simpler components. In our given exercise, simplifying the trigonometric function using identities simplified our derivative calculation drastically.

Exploring the Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry. It states that for any angle \( x \), the sum of the squares of the sine and cosine of \( x \) is always one. Mathematically, this is written as \( \sin^2 x + \cos^2 x = 1 \).
This identity originates from the well-known Pythagorean theorem in geometry, which relates to the sides of a right triangle. The identity holds true for all real numbers \( x \) and is pivotal in simplifying trigonometric expressions.
In our original function \( f(x) = \sin^{2}x + \cos^{2}x \), the identity allows us to recognize that no matter what value \( x \) takes, \( f(x) \) will always be 1, which gives us a much simpler function to work with for differentiation.

The Nature of Constant Functions

Constant functions are a special case in mathematics. They are functions that return the same value no matter what input is given. In mathematical terms, if \( f(x) = c \) where \( c \) is a constant, then \( f(x) \) is a constant function.
The unique thing about constant functions is their derivatives. No matter how you look at it, the rate of change of a constant is always zero. This makes intuitive sense because a constant has no change—it stays the same.

• For example, in our exercise, once we applied the Pythagorean identity, \( f(x) \) was simplified to 1, making it a constant function.
• Hence, the derivative, \( f'(x) \), becomes \( 0 \).

Understanding this helps in quickly identifying scenarios where derivative calculations can be vastly simplified.