Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\cot x=\frac{\cos x}{\sin x} $$

Step-by-Step Solution

Answer

The derivative of \( y = \cot x \) is \( -\csc^2x \).

Step 1: Understand the Derivative Rule Needed

To find the derivative of the function \( y = \cot x \), we can use the quotient rule since \( \cot x = \frac{\cos x}{\sin x} \). The quotient rule states that if you have a function \( \frac{u}{v} \), then its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \), where \( u \) and \( v \) are differentiable functions.

Step 2: Identify Functions for Quotient Rule

Here, \( u = \cos x \) and \( v = \sin x \). Next, find the derivatives: \( u' = \frac{d}{dx}(\cos x) = -\sin x \) and \( v' = \frac{d}{dx}(\sin x) = \cos x \).

Step 3: Apply the Quotient Rule

Substitute the identified functions and their derivatives into the quotient rule formula: \[\left( \frac{\cos x}{\sin x} \right)' = \frac{\sin x \cdot (-\sin x) - \cos x \cdot \cos x}{\sin^2 x}\] Simplify to get:\[= \frac{-\sin^2 x - \cos^2 x}{\sin^2 x}\].

Step 4: Simplify Using Trigonometric Identity

Use the Pythagorean identity \( \sin^2x + \cos^2x = 1 \) to simplify the expression. The numerator becomes \[-(\sin^2 x + \cos^2 x) = -1\].So, the expression is \[\frac{-1}{\sin^2x} = -\csc^2x\].

Step 5: Conclusion

Thus, the derivative of \( y = \cot x \) with respect to \( x \) is \( \frac{d}{dx}(\cot x) = -\csc^2x \).

Key Concepts

Understanding the Quotient Rule

The quotient rule is a fundamental tool in calculus for finding the derivative of a quotient of two functions. It is particularly useful when you have a function given as a fraction, like \( \cot x = \frac{\cos x}{\sin x} \). The rule itself can be remembered by the formula:

• If you have a function \( \frac{u}{v} \), where \( u \) and \( v \) are differentiable functions of \( x \), the derivative is given by:
• \( (\frac{u}{v})' = \frac{v \cdot u' - u \cdot v'}{v^2} \).

This formula requires you to find the derivatives of both the numerator (\( u \)) and the denominator (\( v \)), and then apply them to the formula. It works by subtracting the product of the denominator and the derivative of the numerator from the product of the numerator and the derivative of the denominator, all divided by the square of the denominator. This may sound complex at first, but it becomes intuitive with practice.
Remember, the quotient rule is all about dealing with fractions of functions and is pivotal in calculus differentiation problems.

Using Trigonometric Identities

Trigonometric identities are like a secret weapon in simplifying calculus expressions. When you're dealing with trigonometric functions like \( \cot x = \frac{\cos x}{\sin x} \), these identities help you simplify and transform expressions to make differentiation easier.

• The Pythagorean identity, which states that \( \sin^2 x + \cos^2 x = 1 \), is particularly important.
• In our exercise, it was used to simplify \(-\sin^2 x - \cos^2 x\) to \(-1\).

By applying this identity, you can transform complex trigonometric expressions into simpler forms that are easier to differentiate or integrate. Always keep a list of key trigonometric identities handy when working on calculus problems, as they can save you a lot of time and effort during problem-solving.

Derivative of the Cotangent Function

The cotangent function, \( \cot x \), is the reciprocal of the tangent function and can be expressed as \( \frac{\cos x}{\sin x} \). To find the derivative of \( \cot x \), we use calculus differentiation rules such as the quotient rule.

• Start by identifying that \( u = \cos x \) and \( v = \sin x \).
• Differentiate each: \( u' = -\sin x \), \( v' = \cos x \).
• Apply these in the quotient rule formula to get the expression \( \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} \).
• Simplifying with the trigonometric identity shows that the derivative is \(-\csc^2 x \), where \( \csc x = \frac{1}{\sin x} \).

This result, \( \frac{d}{dx}(\cot x) = -\csc^2 x \), is an essential derivative to remember and forms the basis for understanding more complex trigonometric derivatives. It highlights the interaction between calculus methods and trigonometric identities.

Approach to Calculus Problem Solving

Solving calculus problems effectively requires a structured approach. Here's a simplified strategy you can apply:

• Understand the problem: Identify what is being asked and recognize which calculus principles apply, such as differentiation using the quotient rule in this exercise.
• Break down the function: Decide which parts of the function need to be treated individually, and apply relevant differentiation rules.
• Use mathematical tools: Utilize trigonometric identities and differentiation rules like the quotient rule to manipulate expressions.
• Simplify the expression: Reduce complex expressions using known formulas or identities. This often makes subsequent steps easier.
• Verify your results: Cross-check your derivative or integral with simpler examples or ask yourself if it logically aligns with the function's behavior.

By following these steps, you can systematically approach and solve calculus problems with greater confidence and efficiency. Remember, practice is key to becoming proficient, and each problem solved adds to your understanding of calculus concepts.