Calculating the Derivative of a Complex Function: Step-by-Step Guide

Introduction

Understanding how to calculate the derivative of a complex function is crucial for various fields, including physics, engineering, and mathematics. This guide will walk you through the process of finding the derivative of a given function using the product rule, chain rule, and derivative formulas for common functions. Specifically, we will detail the steps to find the derivative of:

y x - 1 sqrt{2x - x^2} arcsin x - 1

Step-by-Step Calculation

Step 1: Differentiating the First Term

y x - 1 sqrt{2x - x^2}

Differentiate the first term using the product rule. Let u x - 1 and v sqrt{2x - x^2}. The product rule states: frac{dy}{dx} uv vu'. Find v: v sqrt{2x - x^2} (2x - x^2)^{frac{1}{2}} Use the chain rule to differentiate v: v' frac{1}{2}(2x - x^2)^{-frac{1}{2}} cdot (2 - 2x) v' frac{2 - 2x}{2sqrt{2x - x^2}} frac{1 - x}{sqrt{2x - x^2}} Substitute u and v' into the product rule: frac{d}{dx} [x - 1 sqrt{2x - x^2}] (x - 1)sqrt{2x - x^2} (1)frac{1 - x}{sqrt{2x - x^2}}

Step 2: Differentiating the Second Term

y arcsin x - 1

The derivative of arcsin z is frac{1}{sqrt{1 - z^2}}. Here, z x - 1. frac{d}{dx} [arcsin x - 1] frac{1}{sqrt{1 - (x - 1)^2}} Simplify the expression: frac{1}{sqrt{1 - x^2 2x - 1}} frac{1}{sqrt{2x - x^2}}

Step 3: Combine the Results and Simplify

Now, combine the derivatives of the two terms:

frac{dy}{dx} (x - 1)sqrt{2x - x^2} frac{1 - x}{sqrt{2x - x^2}} frac{1}{sqrt{2x - x^2}} Combine the terms: frac{dy}{dx} sqrt{2x - x^2} frac{x - 1(1 - x)}{sqrt{2x - x^2}} frac{1}{sqrt{2x - x^2}} Simplify: frac{dy}{dx} sqrt{2x - x^2} frac{1 - x - 1}{sqrt{2x - x^2}} Finally, combine the last two terms: frac{dy}{dx} frac{1 - x - 1}{sqrt{2x - x^2}}

Final Result

The derivative of the function is:

frac{dy}{dx} frac{1 - x - 1}{sqrt{2x - x^2}}

This result can be further simplified depending on specific needs.

Additional Notes

Understanding and applying the product rule, chain rule, and derivative formulas for common functions are key to solving complex differentiation problems. Practice and repetition will help in mastering these techniques.