The Reciprocal Rule

b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.

검증된 단계별 안내

Start by recalling the Reciprocal Rule, which states that if you have a function \( f(x) \), the derivative of its reciprocal \( \frac{1}{f(x)} \) is given by \( -\frac{f'(x)}{(f(x))^2} \).

Next, recall the Product Rule for derivatives, which states that if you have two functions \( u(x) \) and \( v(x) \), the derivative of their product \( u(x)v(x) \) is \( u'(x)v(x) + u(x)v'(x) \).

To derive the Quotient Rule, consider a function \( \frac{u(x)}{v(x)} \). This can be rewritten as \( u(x) \cdot \frac{1}{v(x)} \).

Apply the Product Rule to \( u(x) \cdot \frac{1}{v(x)} \). Let \( u(x) \) be the first function and \( \frac{1}{v(x)} \) be the second function. The derivative is \( u'(x) \cdot \frac{1}{v(x)} + u(x) \cdot \left(-\frac{v'(x)}{(v(x))^2}\right) \).

Simplify the expression from the previous step to obtain the Quotient Rule: \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). This shows how the Reciprocal Rule and the Product Rule together imply the Quotient Rule.

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주요 개념

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Reciprocal Rule

The Reciprocal Rule in calculus states that the derivative of the reciprocal of a function f(x), denoted as 1/f(x), is given by -f'(x)/[f(x)]^2. This rule is essential for understanding how changes in the reciprocal of a function relate to changes in the function itself, particularly when combined with other derivative rules.

Derivative Product Rule

The Derivative Product Rule is a fundamental concept in calculus used to find the derivative of the product of two functions. If u(x) and v(x) are differentiable functions, the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x). This rule is crucial for handling derivatives involving products of functions, which is often encountered in complex calculus problems.

Derivative Quotient Rule

The Derivative Quotient Rule provides a method for differentiating a quotient of two functions. If u(x) and v(x) are differentiable functions, the derivative of their quotient u(x)/v(x) is given by [u'(x)v(x) - u(x)v'(x)]/[v(x)]^2. Understanding this rule is vital for solving problems involving division of functions, and it can be derived using the Reciprocal Rule and the Product Rule.

The Reciprocal Ruleb. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.