Derivative of y sin x - cos x / (sin x cos x): A Comprehensive Guide

Understanding the derivatives of trigonometric functions is a critical concept in calculus, with applications in various fields including physics, engineering, and economics. This article delves into the detailed process of finding the derivative of the function y sin x - cos x / (sin x cos x) using multiple approaches. Whether you are a student, a professional, or simply someone with an interest in mathematics, this guide is designed to provide you with a thorough understanding of the topic.

Introduction

The function in question, y sin x - cos x / (sin x cos x), involves both sine and cosine functions. To find its derivative, we can use various methods including the Quotient Rule and the Chain Rule. This guide will demonstrate each approach to solve the problem, highlighting the steps and techniques involved, and explaining how to apply them.

Derivative Using the Quotient Rule

Let's begin by using the Quotient Rule to find the derivative of y sin x - cos x / (sin x cos x).

Step 1: Define u and v as follows: u sin x - cos x v sin x cos x

Step 2: Calculate du/dx and dv/dx:

du/dx cos x sin x dv/dx cos2x - sin2x (using the product rule)

Step 3: Apply the Quotient Rule:

y' (v du/dx - u dv/dx) / v2

Substitute u, v, du/dx, dv/dx into the Quotient Rule formula:

y' [(sin x cos x)(cos x sin x) - (sin x - cos x)(cos2x - sin2x)] / (sin x cos x)2

Simplify the expression:

y' (sin x cos2x sin2x cos x - sin x cos2x sin2x cos x) / (sin x cos x)2

y' (2 sin x cos x) / (sin x cos x)2

Further simplification gives:

y' (2 / (sin x cos x))

Hence, the derivative of y sin x - cos x / (sin x cos x) using the Quotient Rule is:

y' 2 / (sin x cos x)

Alternative Methods

In addition to the Quotient Rule, we can also apply alternative methods to simplify and solve the derivative of the given function.

Method 1: Simplifying the Expression

Before finding the derivative, we can simplify the given function:

y (sin x - cos x) / (sin x cos x)

Express the numerator and denominator in terms of tangent: sin x tan x * cos x cos x 1 / sec x

Substitute and simplify:

y (tan x - 1) / tan x 1

This simplifies to:

y tan(x - π/4)

Therefore, the derivative is:

y' sec2(x - π/4)

Simplify further:

y' 2 / (1 cos(2x - π/2))

Which simplifies to:

y' 2 / (1 - sin(2x))

And finally:

y' 2 / (sin x cos x)2

Method 2: Using Trigonometric Identities

We can also use trigonometric identities to simplify the original function:

y (sin x - cos x) / (sin x cos x)

Rewrite the numerator:

y (sin x / cos x) - (cos x / sin x) tan x - cot x

Using the derivative of tangent and cotangent:

y' sec2x csc2x