Problem 31 - The Calculus with Analytic Geometry

Question

Suppose that \(y\) is a function of \(v, v\) is a function of \(u\), and \(u\) is a function of \(x\), and that the derivatives \(D_{v} y, D_{u} v\), and \(D_{x} u\) all exist. Prove the chain rule for three functions: $$ D_{x} y=\left(D_{v} y\right)\left(D_{u} v\right)\left(D_{x} u\right) $$

Step-by-Step Solution

Answer

The chain rule for three functions is: \[D_{x} y = \left(D_{v} y\right)\left(D_{u} v\right)\left(D_{x} u\right).\]

Step 1: Understanding the Functions

First, identify the relationships: \(y\) is a function of \(v\), \(v\) is a function of \(u\), and \(u\) is a function of \(x\). This can be written as \(y(v(u(x)))\).

Step 2: Applying the Chain Rule Step-by-Step

To find the derivative of \(y\) with respect to \(x\), we need to apply the chain rule in steps. Start by finding the derivative of \(y\) with respect to its immediate variable and proceed step-by-step.

Step 1: Derivative of \(y\) with Respect to \(v\)

Since \(y\) is a function of \(v\), find the derivative of \(y\) with respect to \(v\): \[D_{v} y = \frac{dy}{dv}.\]

Step 2: Derivative of \(v\) with Respect to \(u\)

Next, since \(v\) is a function of \(u\), find the derivative of \(v\) with respect to \(u\): \[D_{u} v = \frac{dv}{du}.\]

Step 3: Derivative of \(u\) with Respect to \(x\)

Finally, since \(u\) is a function of \(x\), find the derivative of \(u\) with respect to \(x\): \[D_{x} u = \frac{du}{dx}.\]

Step 6: Combining the Derivatives

According to the chain rule for three functions, combine the derivatives obtained in steps 1, 2, and 3: \[D_{x} y = (D_{v} y)(D_{u} v)(D_{x} u) = \frac{dy}{dv} \cdot \frac{dv}{du} \cdot \frac{du}{dx}.\]

Step 7: Final Proof

This confirms the chain rule for three functions: \[D_{x} y = \left(D_{v} y\right)\left(D_{u} v\right)\left(D_{x} u\right).\]

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. For a function of one variable, the derivative represents the slope of the tangent line to the function's graph at any given point.
To compute derivatives, we use a process called differentiation. Differentiation involves applying specific rules and techniques, such as the power rule, product rule, quotient rule, and the chain rule, to find the derivative.Understanding derivatives is crucial because they are used in various fields, such as physics, engineering, economics, and biology. Derivatives help us understand rates of change, optimize functions, and solve differential equations.
In our exercise, we are using the chain rule to find the derivative of a composite function. Make sure to remember the following key points about derivatives:

• The derivative of a constant is zero.
• The derivative of a sum is the sum of the derivatives.
• The derivative of a product is found using the product rule.
• The derivative of a quotient is found using the quotient rule.
• The derivative of a composite function, like in our exercise, is found using the chain rule.

Composite Functions

A composite function is created when one function is applied to the result of another function. In our exercise, we have a composite function where one function is inside another:

• Function y depends on v
• Function v depends on u
• Function u depends on x

This can be written as y(v(u(x))). To differentiate composite functions, we use the chain rule.
The chain rule allows us to take the derivative of a composite function by relating the rate of change of the outer function to the rate of change of the inner function. In its simplest form, the chain rule states: \[\begin{equation}\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}ewline\end{equation}\].For our three-function composite, it becomes:\[\begin{equation}D_{x} y=\frac{dy}{dv}\times\frac{dv}{du}\times\frac{du}{dx}ewline\end{equation}\].It's essential to identify the dependencies clearly and differentiate step-by-step when dealing with composite functions. This approach helps avoid confusion and ensures accuracy.

Proofs

Proofs play a significant role in mathematics. They help verify that certain statements or formulas are indeed valid.In our exercise, we aim to prove the chain rule for three functions.Here's a step-by-step breakdown:

• First, identify the relationships between the functions.
• Next, apply the derivative step-by-step to each function.
• Finally, combine the derivatives to form the chain rule.

Each step is crucial and builds upon the previous one. The first step establishes the relationships: \[\begin{equation}y is a function of v, v is a function of u, and u is a function of x\end{equation}\]The second step involves differentiating each layer. By recognizing that y depends on v, v on u, and u on x, we ensure every derivative is accounted for.Lastly, combining the derivatives confirms our proof: \[\begin{equation}D_{x} y=\frac{dy}{dv}\times\frac{dv}{du}\times\frac{du}{dx}\end{equation}\]This verification process may seem lengthy, but each element ensures clarity and accuracy. Proofs provide the certainty that mathematical statements or formulas consistently hold true under defined conditions. This approach reinforces the importance of structured logic and careful reasoning in mathematics.