Question

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ g(x)=\frac{\sin x}{x^{2}} $$

Short answer

The derivative of the function \( g(x) = \frac{\sin x}{x^{2}} \) is \( \frac{x. \cos x - 2. \sin x}{x^{3}} \).

Step-by-step solution

Identify the Numerator and Denominator

In this function, \( \sin x \) is the numerator 'u' and \(x^{2}\) is the denominator 'v'.

Calculate the Derivative of u and v

The derivative of \( \sin x \) is \( \cos x \) and the derivative of \(x^{2}\) is \(2x\). Let's denote the derivative of u as u' and the derivative of v as v'.

Apply the Quotient Rule

According to the quotient rule, derivative of function \( g(x) \) is \( \frac{v.u' - u.v'}{v^{2}} \). Substituting the expressions for u, v, u' and v', the derivative of \( g(x) \) is \( \frac{x^{2} . \cos x - \sin x . 2x}{(x^{2})^{2}} \). Hence, the derivative of function \( g(x) \) is \( \frac{x . \cos x - 2 . \sin x}{x^{3}} \).

Key concepts

Numerator and Denominator

In calculus, understanding the roles of the numerator and denominator is key when working with functions that involve fractions. Take the function \( g(x) = \frac{\sin x}{x^{2}} \). Here, the numerator is \( \sin x \), referred to as \( u \), and the denominator is \( x^{2} \), referred to as \( v \).

• The numerator \( u = \sin x \) represents the function in the top part of the fraction.
• The denominator \( v = x^{2} \) represents the function in the bottom part of the fraction.

Identifying these components helps streamline the process of differentiation, especially when applying the Quotient Rule, which is essential for tackling derivatives involving fractions.

Derivative of function

The derivative of a function captures how the function changes at any given point. It's like mapping the slope or the rate of change.For \( g(x) = \frac{\sin x}{x^{2}} \), we need to find \( g'(x) \), the derivative. Instead of differentiating directly, which can be tricky given the fraction, the Quotient Rule comes handy.

• To use the Quotient Rule, calculate the derivative of the numerator, \( u' = \cos x \).
• Calculate the derivative of the denominator, \( v' = 2x \).

These derivatives act as building blocks for using the Quotient Rule and finding the overall derivative of the given function.

Trigonometric derivatives

Trigonometric functions, such as \( \sin x \) and \( \cos x \), have specific rules for differentiation. Knowing these can greatly simplify problems.

• The derivative of \( \sin x \) is \( \cos x \).
• These derivatives are used frequently, not just in pure trigonometry but in calculus where trigonometric expressions are embedded within broader functions.

For the function \( g(x) = \frac{\sin x}{x^{2}} \), using \( \cos x \) as the derivative of \( \sin x \) is crucial for finding \( g'(x) \) correctly.

Derivatives with fractions

Handling derivatives of fractions is where the Quotient Rule shines. It provides a structured way to differentiate functions in the form \( \frac{u}{v} \).The rule itself states: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^{2}} \]For \( g(x) = \frac{\sin x}{x^{2}} \), you apply the Quotient Rule as follows:

• Plug in \( u = \sin x \), \( u' = \cos x \), \( v = x^{2} \), \( v' = 2x \) into the formula.
• Compute \( x^{2} \cdot \cos x - \sin x \cdot 2x \).
• Divide by \( (x^{2})^{2} = x^{4} \).
• The result simplifies to \( \frac{x \cdot \cos x - 2 \cdot \sin x}{x^{3}} \).

Understanding this method ensures you can handle any similar derivative task involving fractions and trigonometric functions.