Question

In Problems 5-38, differentiate. $y=\frac{x^2+6}{\sqrt{x^2+5}}$

   In Problems 5-38, differentiate.

$y=\frac{x^2+6}{\sqrt{x^2+5}}$

Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences, Global Edition

Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences, Global Edition

Ernest Haeussler,… 11th Edition

Chapter 11, Problem 33

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Here, the function is \( y = \frac{x^2 + 6}{\sqrt{x^2 + 5}} \). This is a quotient of two functions, so we will use the quotient rule for differentiation. Show more…

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In Problems 5-38, differentiate. $y=\frac{x^2+6}{\sqrt{x^2+5}}$

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. This concept is fundamental in calculus for analyzing the behavior of functions.

Quotient Rule

The quotient rule is a method used for differentiating functions that are expressed as the ratio of two differentiable functions. It states that the derivative of a quotient f(x)/g(x) is given by (g(x)f'(x) - f(x)g'(x)) divided by the square of g(x), providing a systematic way to handle complex rational expressions.

Chain Rule

The chain rule is used to differentiate composite functions, where one function is applied inside another. This rule states that the derivative of the composite function f(g(x)) is the derivative of f evaluated at g(x) multiplied by the derivative of g(x), allowing one to break down complex expressions into simpler parts.

Power Rule

The power rule is a basic differentiation rule that applies to functions of the form x^n, where n is any real number. It asserts that the derivative of x^n is n multiplied by x raised to the power of n-1, and is a foundational tool for differentiating polynomial expressions.

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