Question

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

Short answer

$\frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{dy}{dx}$.

Step-by-step solution

Understand the given functions

Given the functions, there are two dependencies: \ 1. The function $z$ depends on both $x$ and $y$, i.e., $z=z(x,y)$. \ 2. The function $y$ depends only on $x$, i.e., $y=y(x)$.

Identify what we need to find

We need to find the derivative of $z$ with respect to $x$, which is represented as $\frac{dz}{dx}$. This requires applying the chain rule.

Apply the chain rule

According to the multivariable chain rule, if $z$ is a function of both $x$ and $y$, and $y$ itself is a function of $x$, the total derivative of $z$ with respect to $x$ is: \ $$\frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{dy}{dx}.$$

Differentiate $z$ partially with respect to $x$

First, compute the partial derivative of $z$ with respect to $x$: \ $$\frac{\partial z}{\partial x}.$$

Differentiate $z$ partially with respect to $y$

Next, compute the partial derivative of $z$ with respect to $y$: \ $$\frac{\partial z}{\partial y}.$$

Differentiate $y$ with respect to $x$

Finally, compute the derivative of $y$ with respect to $x$: \ $$\frac{dy}{dx}.$$

Combine the results

Combine all the partial derivatives and the derivative found in steps 4 to 6 using the chain rule formula: \ $$\frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{dy}{dx}.$$

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. When dealing with functions of several variables, the chain rule helps us understand how a change in one variable affects another variable. Let's dive into how the chain rule applies in multivariable calculus.

In our example, we are given two functions: $z=z(x,y)$ where $z$ depends on both $x$ and $y$, and $y=y(x)$ where $y$ depends on $x$. This means that any change in $x$ will result in changes in both $z$ and $y$.

Using the chain rule for multivariable functions, we can find the total derivative of $z$ with respect to $x$. The formula is given by:

$\frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{dy}{dx}$

This means that the total change in $z$ with respect to $x$ is the sum of the direct change of $z$ with respect to $x$ and the change of $z$ through $y$, which in turn depends on $x$.

Partial Derivatives

Partial derivatives are used to find the rates of change of multivariable functions with respect to one variable while keeping the other variables constant. In our example, we need to compute the partial derivatives of $z$ with respect to $x$ and $y$.

First, the partial derivative of $z$ with respect to $x$ is noted as $\frac{\partial z}{\partial x}$. This represents the rate of change of $z$ as $x$ changes, holding $y$ constant.

Next, the partial derivative of $z$ with respect to $y$ is $\frac{\partial z}{\partial y}$. This represents the rate of change of $z$ as $y$ changes, while keeping $x$ constant.

These partial derivatives are essential in applying the chain rule because they help us understand how $z$ is influenced directly by each variable individually. Once we have these partial derivatives, we can plug them into the chain rule formula to find the total derivative.

The actual calculations of these partial derivatives depend on the specific form of the function $z(x,y)$, which is not detailed in the given exercise but is crucial in any practical application of the concept.

Total Derivative

The total derivative measures how a function of several variables changes as the input variables change. In our context, the total derivative of $z$ with respect to $x$ captures how $z$ changes not only directly due to $x$, but also indirectly through $y$, which is itself a function of $x$.

Using the chain rule, we find the total derivative through the formula:

$\frac{dz}{dx}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}\frac{dy}{dx}$

Here, $\frac{\partial z}{\partial x}$ is the partial derivative of $z$ with respect to $x$, accounting for the direct change in $z$ as $x$ varies. The term $\frac{\partial z}{\partial y}\frac{dy}{dx}$ represents the indirect change in $z$ due to the change in $y$ (which changes because $x$ changes).

In simpler terms, the total derivative tells us the overall rate of change of $z$ when $x$ changes, considering both its immediate impact on $z$ and its effect through $y$. This holistic approach is what makes the total derivative a powerful tool in understanding multivariable functions.