Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ \frac{d}{d z}\left(\frac{1}{(F(z))^{2}}\right) $$

Step-by-Step Solution

Answer

The derivative is \( -\frac{2F'(z)}{(F(z))^3} \).

Step 1: Identify the function for differentiation

We need to differentiate \( \frac{1}{(F(z))^2} \) with respect to \( z \). Notice that the function is of the form \( (F(z))^{-2} \).

Step 2: Apply the power rule

According to the power rule, if \( g(z) = (F(z))^{-2} \), then its derivative \( g'(z) \) is \( -2(F(z))^{-3} \cdot F'(z) \).

Step 3: Differentiate the outer function

Applying the power rule, we find the derivative of the outer function which is \( -2(F(z))^{-3} \).

Step 4: Differentiate the inner function

The derivative of the inner function, \( F(z) \), is \( F'(z) \).

Step 5: Combine results using the chain rule

Using the chain rule: \( g'(z) = -2(F(z))^{-3} \cdot F'(z) \).

Step 6: Express the final result

Thus, the derivative of \( \frac{1}{(F(z))^2} \) with respect to \( z \) is \[ \frac{d}{d z}\left(\frac{1}{(F(z))^{2}}\right) = -\frac{2F'(z)}{(F(z))^3}. \]

Key Concepts

Chain Rule

The chain rule is essential when dealing with composite functions. A composite function occurs when one function is applied inside another, like nesting boxes within boxes. In the context of differentiation, imagine having an onion where you need to peel off each layer one by one. The outer layer is differentiated first, and then each inner layer follows suit.

To use the chain rule, identify both the outer and inner functions. The rule states that to find the derivative of the outer function multiplied by the derivative of the inner function, follow this formula: if we have a function \( h(x) = f(g(x)) \) then the derivative is given by \( h'(x) = f'(g(x)) \cdot g'(x) \).

This explains why in the original problem, after applying the power rule to the outer function \( (F(z))^{-2} \), we then needed to differentiate the inner function \( F(z) \) and combine them.

Power Rule

The power rule is one of the most frequently used rules in calculus, especially for polynomials and expressions where a variable is raised to a constant power. It simplifies the process of finding the derivative significantly. The power rule states that if you have a function of the form \( x^n \), its derivative will be \( nx^{n-1} \).

However, it's equally applicable for functions like \( (F(z))^n \), where \( n \) is a constant. In our exercise, the expression \( (F(z))^{-2} \) uses the power rule to find its derivative. By bringing down the exponent, and subtracting one from it, we get \( -2(F(z))^{-3} \). This was the first part of solving the problem, before applying the chain rule to incorporate the necessary derivative of the inner function.

Derivatives

Derivatives are a core concept in calculus and are fundamental in understanding how a function changes. At its heart, a derivative represents the rate of change of a function concerning its variable. If you think of driving a car, the derivative is analogous to the speedometer, showing how quickly your location is changing with time.

When differentiating a function, you aim to find the slope or the steepness of the function at any point. This is crucial for finding maximums, minimums, and points of inflection, and is extensively used across sciences.

In the given exercise, the derivative helps us express how the reciprocal of the square of \( F(z) \) changes as \( z \) changes. The final result showed the combination of methods where both the power rule and the chain rule were applied to find the derivative of the composite function.