Problem 80 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y\) is a differentiable function of \(u, u\) is a differentiable function of \(v\), and \(v\) is a differentiable function of \(x\), then \(\frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d v} \cdot \frac{d v}{d x}\)

Step-by-Step Solution

Answer

The given statement is true because it correctly represents the chain rule for differentiating compositions of three functions: \(y(u(v(x)))\).

Step 1: Understanding the Chain Rule

Chain rule in calculus is a rule for differentiating compositions of functions. In its simplest single-variable form, if we have two functions \(y(u)\) and \(u(v)\) that are both differentiable, then the derivative of the composite function \(y(u(v))\) with respect to \(v\) is the product of the derivatives of \(y\) and \(u\) as follows: \(\frac{dy}{dv} = \frac{dy}{du} \cdot \frac{du}{dv}\)

Step 2: Extend the Chain Rule

When differentiating a function which is a composition of three or more functions, the chain rule still applies. The derivative of \(y(u(v(x)))\) with respect to \(x\) is calculated as the product of the derivatives of \(y, u, v\) as follows: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{dx}\)

Step 3: Compare with the Given Statement

Comparing the resulting formula from step 2 with the given statement, we see that they are identical; thus, the statement is true.

Key Concepts

Differentiable Function

Understanding the concept of a differentiable function is crucial in calculus. A function is considered differentiable at a point if it has a derivative at that point. This means that there exists a tangent to the curve of the function at that particular point, and the function is smooth (without any sharp corners or cusp) around that area.

Key Characteristics of Differentiable Functions:

• A differentiable function is continuous; however, a continuous function is not necessarily differentiable.
• The derivative of a differentiable function represents the rate of change of the function's output with respect to its input.
• Graphically, if you can draw a tangent at every point along the function's curve without lifting your pencil, the function is likely differentiable everywhere within that range.

The concept of differentiability forms the backbone of more complex topics in calculus, such as optimization and the analysis of function behavior.

Derivative of Composite Functions

The derivative of composite functions is found using the chain rule, one of the most important and powerful tools in calculus. When dealing with composite functions, one function is nestled inside another, like layers of an onion. The chain rule allows us to take the derivative of these nested functions by 'peeling away' each layer at a time.

Applying the Chain Rule:

• First, identify the outer and inner functions.
• Differentiate the outer function while keeping the inner part unchanged.
• Then, multiply this result by the derivative of the inner function.

For functions composed of more than two layers, you apply the chain rule repeatedly, working from the outermost layer to the innermost layer, multiplying the derivatives as you go. This systematic process breaks down more complex differentiations into simpler steps.

Calculus

The field of calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two primary branches: differential calculus and integral calculus.

Differential calculus is concerned with the concept of a derivative, which measures how a function changes as its input changes. Integral calculus, on the other hand, deals with the concept of an integral, which measures the accumulation of quantities and is, in a sense, the inverse operation of differentiation.

Applications of Calculus:

• Calculating slopes of curves and finding the rate of change at any given point.
• Maximizing or minimizing physical quantities like distance or energy.
• Understanding and predicting the behavior of physical systems over time, such as in physics or engineering.

Calculus is an essential tool in various fields such as science, engineering, economics, statistics, and many more, allowing for the modeling of complex systems and the solution of problems that can not be approached with algebra alone.