Question

Given \(y=(16 x+3)^{-2},\) use the chain rule to find \(d y / d x\). Then rewrite the function as \(y=1 /(16 x+3)^{2}\) and firid \(d y / d x\) by the quotient rule. Are the answers identical?

Short answer

Yes, both methods give the derivative \(-32(16x + 3)^{-3}\), confirming the answers are identical.

Step-by-step solution

Identify the Outer and Inner Functions for Chain Rule

The given function is expressed as \( y = (16x + 3)^{-2} \). Here, the outer function is \( u^{-2} \), where \( u = 16x + 3 \). The inner function is \( u = 16x + 3 \). Identify these properly before applying the chain rule.

Differentiate the Outer Function Using Chain Rule

Differentiate the outer function with respect to the inner function \( u \). The derivative of \( u^{-2} \) with respect to \( u \) is \(-2u^{-3} \).

Differentiate the Inner Function

Differentiate the inner function \( u = 16x + 3 \) with respect to \( x \). The derivative is \( 16 \).

Apply Chain Rule

Using the chain rule formula \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \), we substitute \( \frac{dy}{du} = -2(16x + 3)^{-3} \) and \( \frac{du}{dx} = 16 \) to get \[ \frac{dy}{dx} = -2(16x + 3)^{-3} \cdot 16 = -32(16x + 3)^{-3} \].

Rewriting Function for the Quotient Rule

Rewrite the given function as \( y = \frac{1}{(16x + 3)^{2}} \). This will allow us to use the quotient rule to find the derivative.

Apply Quotient Rule

The quotient rule states \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \). Here, \( u = 1 \) and \( v = (16x + 3)^2 \). Therefore, \( \frac{du}{dx} = 0 \) and \( \frac{dv}{dx} = 2(16x + 3)(16) \).

Compute Derivative Using Quotient Rule

Substitute into the quotient rule formula: \[ \frac{dy}{dx} = \frac{(16x+3)^2 \cdot 0 - 1 \cdot 2(16x + 3)(16)}{((16x + 3)^{2})^2} = \frac{-32(16x + 3)}{(16x + 3)^4} = -32(16x + 3)^{-3} \].

Compare Results

The derivative found using both methods is \( -32(16x + 3)^{-3} \). Thus, both methods yield identical results.

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions, which are functions composed of other functions. It's like peeling layers off an onion. You work from the outside in, focusing on one function at a time. Here's how it works:

Consider a function, where you have an outer function and an inner function. For example, if we have a function expressed as \( y = (f(g(x))) \), the chain rule states that the derivative \( \frac{dy}{dx} \) is obtained by multiplying the derivative of the outer function \( f \) with respect to its inner function \( g \), by the derivative of the inner function \( g \) with respect to \( x \). Hence, the formula for the chain rule is:

\[ \frac{dy}{dx} = \frac{d}{dg}(f(g)) \cdot \frac{dg}{dx} \]

In simpler terms, you first differentiate the outer function as if the inner function was simply a variable, then multiply that result by the derivative of the inner function. This is how you "chain" the derivatives together.

Quotient Rule

The quotient rule is another powerful tool in calculus used to find the derivative of a function that is the quotient of two other functions. Essentially, if you have a function expressed as \( \frac{u}{v} \), where both \( u \) and \( v \) are differentiable functions of \( x \), the quotient rule can be applied.

The formula for the quotient rule is:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

This rule is a bit more involved than the product rule because it deals with the intricate interplay of division. The essence of the quotient rule is that you first differentiate the "top" function (numerator) and the "bottom" function (denominator) separately. Then, you apply the formula, which involves subtracting the product of the derivative of the denominator and the original numerator from the product of the derivative of the numerator and the original denominator. Finally, divide everything by the square of the original denominator.

Mathematical Functions

In mathematics, functions are a way to express a relationship between inputs and outputs. A function takes an input, processes it, and produces an output. Think of it as a machine that takes in a number, performs a specified operation, and spits out a result.

There are many types of functions:

• Linear Functions: These are polynomials of degree one and have the form \( f(x) = mx + b \).
• Quadratic Functions: Polynomials of degree two, generally expressed as \( ax^2 + bx + c \).
• Exponential Functions: These functions have variables in the exponent, such as \( a^x \).
• Rational Functions: Zfunctions represented as the ratio of two polynomials, such as \( \frac{p(x)}{q(x)} \).

Functions can be combined and manipulated in various ways to form composite functions, which is where rules like the chain and quotient rules are handy for differentiation.

Calculus

Calculus is a branch of mathematics that deals with change. It's the mathematics of motion, rates of change, and has a powerful way of analyzing situations where variables are continuously changing. Calculus is essentially divided into two main branches:

• Differential Calculus: This part focuses on understanding and finding the rate of change – the derivative.
• Integral Calculus: This part deals with accumulation of quantities, like areas under curves, and is the reverse process of differentiation.

In calculus, derivatives help us understand how a function changes at any point, moving toward the idea of the slope of a curve at that point. Meanwhile, integrals help us understand the total accumulation of something, like finding the area under a curve or the distance traveled over time. Both aspects of calculus are instrumental in solving problems in physics, engineering, economics, and beyond.