How to Differentiate y 3x^2 √7x 1: A Comprehensive Guide

When dealing with the mathematical function y 3x^2 √7x 1, it is important to understand how to calculate its derivative using proper techniques from calculus.

Understanding the Notation and Assumptions

I will assume you have a basic understanding of several mathematical concepts, including the derivative addition rule and the power rule for differentiation. It is also assumed that you are familiar with the chain rule when dealing with composite functions, such as the square root.

Differentiating the Given Function

Given the function y 3x^2 √7x 1, this can be broken down into three simpler functions: 3x^2, √7x, and 1. Each of these can be differentiated individually and then summed to find the overall derivative. Let's go through the steps one by one.

Step 1: Differentiate 3x^2

Using the power rule, the differentiation of 3x^2 is straightforward:

dy/dx(3x^2) 6x

Step 2: Differentiate √7x

When dealing with the term √7x, it is important to note that the square root can be represented as a fractional exponent. Here, √7x can be rewritten as (7x)^(1/2). We can now differentiate it using the chain rule.

Let u 7x, then z u^(1/2).

dz/du 1/2 * u^(-1/2)

du/dx 7

Applying the chain rule:

dz/dx (1/2 * u^(-1/2)) * 7 (7/2) * (7x)^(-1/2)

Thus, the derivative of √7x is:

dy/dx(√7x) (7/2) * (7x)^(-1/2)

Step 3: Differentiate 1

The derivative of a constant (1) is 0:

dy/dx(1) 0

Summing the Derivatives

To find the derivative of the entire function, we simply sum the derivatives of the individual components:

dy/dx(y) 6x (7/2) * (7x)^(-1/2) 0

Therefore, the derivative of y 3x^2 √7x 1 is:

dy/dx 6x (7/2) * (7x)^(-1/2)

A Note on Ambiguity

There is some ambiguity in the original question, as it could be interpreted in two ways:

y 3x^2 √(7x)

y 3x^2 √(7) * x

For the first interpretation, the derivative is:

dy/dx 6x (7/2) * (7x)^(-1/2)

For the second interpretation, the derivative simplifies to:

dy/dx 6x √7

Summary

In summary, the process of finding the derivative of a complex function involves breaking it down into simpler parts, differentiating each part, and then summing the results. This approach is particularly useful in calculus for handling various types of functions.

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