Question

Let \(z\) be a function of \(x\) and \(y,\) while \(x\) and \(y\) are functions of \(t\) Explain how to find \(d z / d t\).

Short answer

Question: Find the derivative of \(z\) with respect to \(t\), given that \(z\) is a function of \(x\) and \(y\), and both \(x\) and \(y\) are functions of \(t\). Answer: Based on the provided information, the derivative of \(z\) with respect to \(t\) can be found using the Chain Rule: \(\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}\). To find the specific expression for \(\frac{dz}{dt}\), you will need to compute the necessary derivatives: \(\frac{\partial z}{\partial x}\), \(\frac{\partial z}{\partial y}\), \(\frac{dx}{dt}\), and \(\frac{dy}{dt}\) based on the given functions \(z(x,y)\), \(x(t)\), and \(y(t)\).

Step-by-step solution

Write down the given functions

We are given that \(z\) is a function of \(x\) and \(y\), which can be written as \(z(x, y)\). Moreover, \(x\) and \(y\) are functions of \(t\), which can be written as \(x(t)\) and \(y(t)\), respectively.

Apply the Chain Rule for \(\frac{dz}{dt}\)

To find \(\frac{dz}{dt}\), we need to apply the Chain Rule twice because \(z\) depends on \(x\) and \(y\), and both of them depend on \(t\). So, we get: \(\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}\) Here, \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) are the partial derivatives of \(z\) with respect to \(x\) and \(y\), respectively, and \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) are the derivatives of \(x\) and \(y\) with respect to \(t\), respectively.

Compute the necessary derivatives

We need to compute the following four derivatives: \(\frac{\partial z}{\partial x}\), \(\frac{\partial z}{\partial y}\), \(\frac{dx}{dt}\), and \(\frac{dy}{dt}\). These derivatives will be determined by the specific forms of the functions \(z(x,y)\), \(x(t)\) and \(y(t)\), which were not provided in the exercise. Once you find these derivatives based on the given functions, you can proceed to the next step.

Substitute the derivatives into the Chain Rule formula

Substitute the calculated derivatives from Step 3 into the formula from Step 2 to find the expression for \(\frac{dz}{dt}\): \(\frac{dz}{dt} = \frac{\partial z}{\partial x}\cdot\frac{dx}{dt} + \frac{\partial z}{\partial y}\cdot\frac{dy}{dt}\) This final expression will give you the derivative of \(z\) with respect to \(t\).

Key concepts

Partial Derivatives

The concept of partial derivatives is imperative when dealing with functions of several variables. In the scenario where we have a function like z that depends on two variables, x and y, To fully understand this, imagine you are in a hilly terrain, represented by the function z, where x and y give your position on a map. If you're interested in how the height changes as you walk directly east, that's where you'd use the partial derivative of z with respect to x, written as \( \frac{\partial z}{\partial x} \). Similarly, to know how the height changes going directly north, you'd use \( \frac{\partial z}{\partial y} \).

It's like asking how much the terrain inclines or declines if you move in just one direction at a time while keeping the other direction fixed. Finding partial derivatives entails treating all other variables as constants and differentiating with respect to the variable of interest.

Multivariable Calculus

Multivariable calculus expands calculus to functions of more than one variable, like our function z that depends on both x and y. While single-variable calculus asks questions about functions that change with respect to one independent variable, multivariable calculus addresses functions involving several changing entities.

In practice, this means we have to consider how changing each variable affects the outcome – much like how both the grade and direction of a slope alter the difficulty of a hike. The chain rule we use in multivariable scenarios is a powerful tool that combines derivatives based on how each individual variable affects the function when all are changing simultaneously.

Derivative with Respect to Time

When we say 'derivative with respect to time', we're often concerned with how a quantity changes as time progresses. In real-life contexts, we might be tracking the speed of a car (when distance changes over time) or the decay of a radioactive substance (when mass changes over time).

In the given problem, we're interested in how z changes over time while acknowledging that z's dependency on x and y is indirect; they're the middlemen that themselves change with time. The chain rule elegantly stitches this relationship together, essentially saying: to know how z changes with time, find out how z changes with x and y, and then see how both x and y change with time.

Function Differentiation

Function differentiation is the process of finding the derivative of a function. In simpler terms, it's like figuring out the rate at which something is changing at any given moment. For a function of a single variable, this is relatively straightforward – it's the slope of the tangent line to the function's graph at any point.

But in cases where a function depends on more than one variable, differentiation gets more nuanced. We use partial derivatives to investigate the rate of change along each axis. Then, with the chain rule, we can weave these rates together when the change is influenced by several variables that are themselves changing, potentially even with time. It's a dance of dependency that requires careful choreography, which mathematics provides us through the principles of calculus.