Problem 193 - Calculus Volume 1

Question

Show that \(y=\csc ^{2} x\) and \(y=\cot ^{2} x\) have the same derivative. What can you say about \(y=\csc ^{2} x-\cot ^{2} x ?\)

Step-by-Step Solution

Answer

The derivatives of both functions are identical. \( y = \csc^2 x - \cot^2 x \) is a constant value of 1.

Step 1: Recall the Differentiation Rules

We need to differentiate the expressions for both \(y = \csc^2 x\) and \(y = \cot^2 x\). Recall that the derivative of \( \csc x\) is \( -\csc x \cot x \) and the derivative of \( \cot x \) is \( -\csc^2 x \).

Step 2: Differentiate \(y = \csc^2 x\)

Using the chain rule, the derivative of \(y = \csc^2 x\) is calculated as follows:\[\frac{d}{dx}(\csc^2 x) = 2 \csc x (-\csc x \cot x) = -2 \csc^2 x \cot x.\]

Step 3: Differentiate \(y = \cot^2 x\)

Again, using the chain rule, differentiate \( y = \cot^2 x \):\[\frac{d}{dx}(\cot^2 x) = 2 \cot x (-\csc^2 x) = -2 \cot x \csc^2 x.\]

Step 4: Compare the Derivatives

Both derivatives calculated in Step 2 and Step 3 result in the expression:\[-2 \csc^2 x \cot x = -2 \cot x \csc^2 x.\]Hence, the derivatives are identical.

Step 5: Analyze \(y = \csc^2 x - \cot^2 x\)

The expression \( y = \csc^2 x - \cot^2 x \) is a known trigonometric identity equal to 1. Therefore, the derivative with respect to \(x\) is 0, as the derivative of a constant is zero.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on change and motion. It helps in finding the rate at which things change, known as a derivative. The derivative tells us how a function behaves as its input changes, giving crucial insights into the dynamics of varying systems.
Understanding differentiation, which is a foundational tool in calculus, involves simple rules that allow us to break down complex functions into simpler components.

• The Product Rule and Quotient Rule help when dealing with multiplication and division of functions.
• The Chain Rule is particularly useful for composite functions, where one function is nested inside another.
• Knowing derivatives of standard functions like sine, cosine, and exponential functions is crucial.
• Differentiation can also be used in real-world applications such as calculating speed in physics or finding maxima and minima in economics.

Calculus and its techniques are a gateway to more advanced topics and their applications in various scientific fields, making it an essential part of foundational mathematics.
Automating differentiation by using these rules simplifies solving complex problems, like the ones involved in this exercise.

Trigonometric identities

Trigonometric identities are equations involving trigonometric functions that hold true for all angles. They are essential tools in simplifying and solving trigonometric expressions.

• The basic identities include \( \sin^2 x + \cos^2 x = 1 \) and \( \tan^2 x + 1 = \sec^2 x\).
• Another important identity is \( \csc^2 x - \cot^2 x = 1\), used in this exercise.

These identities allow us to express complex trigonometric expressions in simpler forms, facilitating easier differentiation and integration.
Understanding and applying these identities also help eliminate redundancy when dealing with similar trigonometric forms, as used in proving that the derivatives in the exercise are equal.
They serve as key steps in verifying trigonometric equations and solving trigonometric equations, crucial for trigonometry-focused calculations.

Chain rule

The chain rule is a fundamental differentiation technique in calculus used to differentiate composite functions.
A composite function is where one function is nested inside another, such as \( y = ( ext{inner function})^2\).
For instance, in differentiating \( y = ext{outer function}( ext{inner function}(x))\), the chain rule allows us to handle each part separately, making complex calculations manageable.
Here's how it works:

• First, differentiate the outer function.
  This gives: \( rac{d}{d( ext{inner})}( ext{outer})\)


• Then, multiply by the derivative of the inner function: \( rac{d}{dx}( ext{inner})\)

Using the chain rule, differentiation tasks like those in this exercise become systematic and less error-prone.
For example, differentiating \( rac{d}{dx}( ext{sin}^2(x))\) involves differentiating the outer function \( ext{sin}(x)^2\) to get \( 2 ext{sin}(x) ext{cos}(x)\) and then multiplying by the derivative of \( ext{sin}(x)\) which is \( ext{cos}(x)\).
Mastering the chain rule is essential for tackling advanced calculus problems, where nested functions frequently appear.