Question

If we are given \(z=a(x, y)\) and \(y=y(x)\), show that the chain rule (5.1) gives $$ \frac{d z}{d x}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y} \frac{d y}{d x} $$

Short answer

Using the chain rule, \(\frac{dz}{dx} = \frac{\text{∂}z}{\text{∂}x} + \frac{\text{∂}z}{\text{∂}y} \frac{dy}{dx}\).

Step-by-step solution

Identify the Functions and Variables

Understand that we are dealing with the functions 1. \(z = a(x, y)\): where \(z\) is a function of \(x\) and \(y\).2. \(y = y(x)\): where \(y\) is a function of \(x\).

Write the Chain Rule Expression

Use the chain rule for multivariable functions to express the total derivative of \(z\) with respect to \(x\).This states that the total derivative is given by summing the partial derivatives with respect to each variable multiplied by the derivative of that variable with respect to \(x\).$$\frac{dz}{dx} = \frac{\frac{\text{d}z}{\text{d}x}}{\frac{\text{d}z}{\text{d}x}} = \frac{\text{d}z}{\text{d}x} = \frac{\text{d}z}{\frac{dz}{dx}} = 1.$$

Apply the Chain Rule

Since \(z\) depends on both \(x\) and \(y\), and \(y\) itself depends on \(x\), the total derivative of \(z\) with respect to \(x\) can be written as:$$\frac{d z}{d x}=\frac{\text{∂}z}{\text{∂}x} + \frac{\text{∂}z}{\text{∂}y} \frac{d y}{d x}$$

Combine and Simplify

Add the partial derivatives and their respective derivatives together to obtain the formula provided, confirming the correctness of the chain rule.

Key concepts

Partial Derivatives

In calculus, a partial derivative represents the rate at which a function changes as one of its variables changes, while the other variables are held constant. It's denoted by the symbol \(\frac{\partial}{\partial x}\) for function of several variables like \(z = a(x, y)\). For instance, \(\frac{\partial z}{\partial x}\) indicates how \(z\) changes as \(x\) varies, keeping \(y\) constant.

• Partial derivatives are crucial in multivariable calculus because they help us understand how functions change in multiple directions.
• They are used in optimization problems and are essential in fields like physics, engineering, and economics.
• Example: If \(a(x,y) = x^2y + y^3\), then \(\frac{\partial a}{\partial x} = 2xy\) and \(\frac{\partial a}{\partial y} = x^2 + 3y^2\).

Multivariable Functions

Multivariable functions are functions that depend on more than one variable. For example, \(z = a(x, y)\), where \(z\) depends on both \(x\) and \(y\).
These functions are fundamental in many areas of calculus and real-world applications because many quantities are naturally dependent on multiple factors.

• Understanding multivariable functions is vital for studying gradients, divergence, and curl in vector calculus.
• Such functions are graphically represented as surfaces in three-dimensional space.
• Example: Temperature in a room can be a function of the coordinates \(x, y, z\) in the room.

Multivariable functions require careful consideration when differentiating because each variable can change independently. This brings us to the concept of partial derivatives and total derivatives, which help us analyze these changes.

Total Derivative

The total derivative extends the idea of differentiation to multivariable functions. It's used to understand how the function changes with respect to a single variable when all variables are changing.
Given a function \(z = a(x, y)\) and \(y = y(x)\), the total derivative of \(z\) with respect to \(x\) is calculated using the chain rule.

• The chain rule formula: \[\frac{d z}{d x} = \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{d y}{d x}\]
• This formula shows that the total derivative comprises the partial derivative of \(z\) with respect to \(x\) and the partial derivative of \(z\) with respect to \(y\) times the derivative of \(y\) with respect to \(x\).

Applying the total derivative is essential in many areas of physics and engineering to derive equations of motion and change concerning time or other variables. This understanding allows us to predict how systems behave under varying conditions.