Question

Two equivalent forms of the Chain Rule for calculating the derivative of \(y=f(g(x))\) are presented in this section. State both forms.

Short answer

Question: State two equivalent forms of the Chain Rule for calculating the derivative of a composition of two functions, \(y = f(g(x))\). Answer: 1. The first form of the Chain Rule is: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\), where \(y = f(u)\) and \(u = g(x)\). 2. The second form of the Chain Rule is: \(\frac{d(f(g(x)))}{dx} = f'(g(x)) \cdot g'(x)\).

Step-by-step solution

Chain Rule Form 1

The first form of the Chain Rule is given by the following expression: $$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}, $$ where \(y = f(u)\) and \(u = g(x)\). In this form, we first find the derivative of the outer function with respect to the inner function (\(\frac{dy}{du}\)), and then multiply it by the derivative of the inner function with respect to the original variable x (\(\frac{du}{dx}\)).

Chain Rule Form 2

The second form of the Chain Rule is given by the following expression: $$ \frac{d(f(g(x)))}{dx} = f'(g(x)) \cdot g'(x). $$ In this form, we compute the derivative of the outer function evaluated at the inner function (\(f'(g(x))\)), and then multiply it by the derivative of the inner function (\(g'(x)\)).

Key concepts

Derivative

The concept of a derivative refers to the rate at which one quantity changes with respect to another. In the realm of mathematics, and more specifically calculus, derivatives measure how a function's output changes as its input changes. Imagine you are driving a car and your speed is the derivative of your position; it tells you how quickly your position changes over time.

In formal terms, if you have a function, say, f(x), the derivative of this function denoted as f'(x) or \frac{{df}}{{dx}} is the limit of the average rate of change as the interval gets infinitely small. In simpler terms, it's the 'instantaneous' rate of change at a point.

Mathematically represented, the derivative is typically found using the limit definition:

\[ f'(x) = \frac{{d}}{{dx}}f(x) = \text{lim}_{h \to 0} \frac{{f(x+h) - f(x)}}{{h}}. \]

Understanding the derivative is crucial as it lays the groundwork for solving more complex problems involving rates of change in various fields like physics, engineering, and economics.

Differentiation

Differentiation is the process of calculating the derivative. It is the mathematical procedure we employ to determine the rate of change of a variable. Differentiation is a fundamental tool in calculus used to dissect functions into their instantaneous rates of change.

There are various rules of differentiation that simplify this process, such as the power rule, product rule, quotient rule, and the focus of our exercise, the Chain Rule. Each rule provides a method to find the derivative of different kinds of functions—whether they are simple polynomials, fractions, or complex compositions of functions.

When differentiating composite functions—functions made up of two or more functions combined together—the Chain Rule becomes indispensable. It allows us to unravel these layers by differentiating the outer function and the inner function sequentially. In application, differentiation can be used to find things like the slope of a curve at a point, or to optimize functions in order to find maximum and minimum values.

Composite Functions

A composite function is essentially a combination of two or more functions such that the output of one function becomes the input of another. In notation, if we have two functions, f(x) and g(x), their composite would be written as f(g(x)) and pronounced as 'f of g of x'.

For example, if f(x) = x^2 and g(x) = 3x + 2, the composite function f(g(x)) would be f(3x + 2) which simplifies to (3x + 2)^2. These compositions reflect more complex relationships and can describe a wide array of real-world scenarios where one action's result is the starting point for another.

When it comes to derivatives of composite functions, this is where our formula from the chain rule comes into play. The chain rule allows us to differentiate composite functions efficiently, by treating each function in the composition separately. This is extremely practical because complex compositions are common in advanced mathematics and many scientific applications.

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It's divided into two main areas: differential calculus and integral calculus. Differential calculus, which encompasses derivatives and differentiation, is concerned with how things change instantaneously, while integral calculus focuses on accumulation and areas under curves.

Calculus is essential in science and engineering, offering a framework for modeling systems in motion, optimizing processes, and predicting future events. It is a tool that allows us to calculate slopes of curves, areas under graphs, and solve problems involving complex rates of change. Beyond these, calculus helps in determining the behavior of functions and understanding the nature of physical space.

Whether through determining instantaneous velocities, finding tangents to curves, or calculating minimum and maximum points on a graph, calculus provides a profound set of techniques to approach and solve a myriad of problems – symbolizing the language of change used across diverse disciplines.