Problem 90 - Brief Applied Calculus

Question

Find a formula for \(\frac{d}{d x}[f(x)]^{-1}\) by writing it as \(\frac{d}{d x}\left[\frac{1}{f(x)}\right]\) and using the Quotient Rule. Be sure to simplify your answer.

Step-by-Step Solution

Answer

\( \frac{d}{dx}[f(x)]^{-1} = \frac{-f'(x)}{[f(x)]^2} \).

Step 1: Express the Function as a Quotient

To begin, note that \( [f(x)]^{-1} \) is equivalent to \( \frac{1}{f(x)} \). This allows us to prepare the expression for differentiation using the quotient rule.

Step 2: Recall the Quotient Rule

The quotient rule states that if you have a function expressed as \( \frac{u(x)}{v(x)} \), then its derivative is \( \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \). For our function \( \frac{1}{f(x)} \), we identify \( u(x) = 1 \) and \( v(x) = f(x) \).

Step 3: Differentiate Using the Quotient Rule

Apply the quotient rule using \( u(x) = 1 \) and \( v(x) = f(x) \). This gives\[ \frac{d}{dx} \left[ \frac{1}{f(x)} \right] = \frac{f(x) \cdot \frac{d}{dx}[1] - 1 \cdot \frac{d}{dx}[f(x)]}{[f(x)]^2} \].Since the derivative of a constant \(1\) is \(0\), the formula simplifies to:\[ \frac{f(x) \cdot 0 - 1 \cdot f'(x)}{[f(x)]^2} = \frac{-f'(x)}{[f(x)]^2} \].

Step 4: Simplify the Expression

The expression simplifies to:\[ \frac{-f'(x)}{[f(x)]^2} \].This is the derivative of \( [f(x)]^{-1} \) with respect to \( x \).

Key Concepts

Quotient Rule

When dealing with derivatives of quotient functions, the Quotient Rule is an essential tool. Understanding how to apply it effectively can simplify complex expressions.
The Quotient Rule provides a method for finding the derivative of a division of two differentiable functions.
If a function is expressed as \( \frac{u(x)}{v(x)} \), where both functions \( u(x) \) and \( v(x) \) are differentiable, then the derivative of this quotient is given by:
\[ \frac{d}{dx} \left[ \frac{u(x)}{v(x)} \right] = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}. \]
In simple terms, you:

• Multiply the derivative of the numerator \( u'(x) \) by the denominator \( v(x) \).
• Subtract the product of the numerator \( u(x) \) and the derivative of the denominator \( v'(x) \).
• Divide the entire result by \( v(x)^2 \).

The Quotient Rule is vital for correctly calculating derivatives when dealing with fractions of functions.

Mathematical Functions

Mathematical functions are expressions that relate one quantity to another.
When working on derivatives in calculus, understanding the nature of functions is crucial.
A function maps inputs (often called \( x \) values) to respective outputs through a specific rule. For instance, in \( f(x) = x^2 \), if \( x \) is 3, then \( f(x) \) results in 9.
There are several types of functions, including:

• Linear functions: These have constant rates of change, like \( f(x) = 3x + 2 \).
• Polynomial functions: Include terms like \( x^2 \), \( x^3 \), etc., such as \( f(x) = x^3 - 4x + 5 \).
• Trigonometric functions: Functions like sine and cosine that deal with angles.
• Exponential functions: Involving constant bases raised to variable powers, such as \( f(x) = 2^x \).
• Rational functions: Ratios of polynomials, like \( f(x) = \frac{x^2+1}{x-3} \).

Knowing how to express and manipulate these functions allows for efficient application of calculus techniques like differentiation.

Differentiation Techniques

Differentiation is a key concept of calculus focused on finding the rate of change of functions.
It’s used to determine how a function's output changes with respect to its input.
There are several techniques to perform differentiation:

• Power Rule: Useful for any function of the form \( f(x) = x^n \). The derivative is \( nx^{n-1} \).
• Product Rule: If you have two multiplied functions \( u(x) \) and \( v(x) \), the derivative is \( u'(x)v(x) + u(x)v'(x) \).
• Chain Rule: Used for composite functions. If \( y = g(f(x)) \), then the derivative is \( g'(f(x)) \cdot f'(x) \).
• Quotient Rule: As previously mentioned, this is for divisions of functions.

Whenever you differentiate a function, the goal is often to find tangent lines, rates of change, or to properly analyze functional behavior visually or analytically.