Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\sin ^{2} x+\cos ^{2} x $$

Step-by-Step Solution

Answer

The derivative \(D_x y\) is 0.

Step 1: Recognize Trigonometric Identity

The given expression is \(y = \sin^2 x + \cos^2 x\). We recognize that \(\sin^2 x + \cos^2 x = 1\) is a fundamental trigonometric identity.

Step 2: Differentiate Using Constant Rule

The function \(y = 1\) is a constant function since \(\sin^2 x + \cos^2 x\) always equals 1. The derivative of any constant function is zero. Therefore, \(D_x y = 0\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving many calculus problems. These identities express relationships between the basic trigonometric functions: sine, cosine, tangent, etc. One of the most fundamental and widely used identities is \[\sin^2 x + \cos^2 x = 1\]This identity is known as the Pythagorean identity. It is derived from the Pythagorean theorem and helps in simplifying expressions involving squared sine and cosine functions.

• The Pythagorean identity is helpful because it simplifies complex trigonometric expressions.
• It verifies trigonometric equations by reducing them to simpler forms.
• This identity is crucial when dealing with derivatives or integrals in calculus.

Understanding and recognizing these identities helps you quickly identify simplifications before undertaking lengthier calculus processes.

Derivative of Constant Function

The concept of differentiation involves finding how a function changes as its input changes. However, when we deal with constant functions, things are much simpler. A constant function doesn't change, so its derivative is always zero.In the given problem, after applying the trigonometric identity, we are left with the constant function \(y = 1\).Differentiation rules tell us that the derivative of a constant value \(c\) is \(D_x c = 0\). This means the rate at which the constant value changes is zero.When you differentiate:

• Look for any parts of the expression that simplify to constants.
• Recall that the derivative of any constant, like our simplified \(1\), is always 0.
• This applies universally, no matter the input variable.

Understanding this rule helps simplify calculus problems involving constants and confirms that some functions may involve fewer steps than they seem at first glance.

Step-by-Step Calculus Solutions

Step-by-step calculus solutions provide a clear path for solving complex problems. They break down a problem into manageable parts, allowing you to follow the logical progression of thought needed to arrive at the solution. Here's how you can use step-by-step solutions effectively:1. **Identify Recognizable Patterns:** Look for fundamental identities or simplifications, like the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\).2. **Apply Relevant Rules:** Once simplified, apply calculus concepts, such as the derivative of a constant, to find the answer. Remember, \(D_x y = 0\) for constant functions.3. **Cross-Verify:** After solving, cross-verify each step to ensure validity and correctness. This not only helps in understanding the solution but also reinforces learning.By practicing these strategies, you'll enhance your problem-solving skills in calculus. Solutions derived step-by-step can lead to better comprehension and retention of mathematical concepts.