91Ó°ÊÓ

When dealing with functions like \( y = \frac{x}{\sqrt{x^2 + 1}} \), we often use the quotient rule to find the derivative. The quotient rule is essential in calculus whenever you need to differentiate a quotient of two functions. Hence, it helps us understand how the rate of change of one function relative to another yields the overall rate change.

• Formula: If you have two functions \( f(x) \) and \( g(x) \), the quotient rule is given by: \[ \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
• Usage: First, differentiate the numerator \( f(x) \) and denominator \( g(x) \) individually. Then substitute into the formula.

In our exercise, we identified \( f(x) = x \) and \( g(x) = \sqrt{x^2 + 1} \). The derivatives are \( f'(x) = 1 \) and for \( g(x) \), we found \( g'(x) = \frac{x}{\sqrt{x^2 + 1}} \), making the application of the quotient rule straightforward. By substituting these into the rule, we got:\[ y' = \frac{1 \cdot \sqrt{x^2+1} - x \cdot \frac{x}{\sqrt{x^2 +1}}}{(\sqrt{x^2 + 1})^2} \]This derivation leads us to see how functions interact in terms of rates of change.

The chain rule is applied when we have a function composed inside another function. It is a powerful technique used in differentiation, especially when tackling composite functions like \( g(x) = \sqrt{x^2 + 1} \).

• Formula: For functions \( h(x) = u(v(x)) \) where \( u \) and \( v \) are functions of \( x \), the chain rule states: \[ h'(x) = u'(v(x)) \cdot v'(x) \]
• Application: First, derive \( u \) with respect to its inner function \( v \), then multiply by the derivative of \( v \) with respect to \( x \).

In this exercise, to find \( g'(x) \) where \( g(x) = \sqrt{x^2 + 1} \), we treated \( g(x) \) as \( (x^2 + 1)^{1/2} \). Applying the chain rule:- \( u(v(x)) = (v(x))^{1/2} \), so \( u'(v(x)) = \frac{1}{2}(v(x))^{-1/2} \)- \( v(x) = x^2 + 1 \), so \( v'(x) = 2x \)Hence, \( g'(x) = \frac{1}{2\sqrt{x^2 + 1}} \times 2x = \frac{x}{\sqrt{x^2 + 1}} \). This factor became crucial in solving the derivative \( y' \) using the quotient rule.

Calculus is a vital branch of mathematics that enables us to explore changes between values related through functions. It primarily focuses on two basic concepts: derivatives and integrals. In this exercise, we concentrated on finding the derivative of a function using two rules from calculus, the quotient and chain rules.

• Derivatives: They represent the rate of change of a function relative to its variable. In essence, they tell us how fast or slow the change happens.
• Rules of Differentiation: Rules like the quotient and chain rules are tools that simplify the process of finding derivatives of complex functions.

Through calculus, complicated functions can be simplified, allowing us to analyze and comprehend the behavior of mathematical models. In our given function \( y = \frac{x}{\sqrt{x^2 + 1}} \), we used calculus techniques to derive its rate of change, demonstrating both the beauty and power of calculus. With practice, these rules become intuitive, enabling you to navigate through sophisticated mathematical challenges effectively.