91Ó°ÊÓ

Partial derivatives are essential when dealing with functions of multiple variables. They enable us to understand how a function changes as one specific variable changes, while the others remain constant. In this exercise, we are dealing with a function of variables, i.e.,

• x represents one dimension, and the other two dimensions are represented by y and z.
• This is where partial derivatives become important.

To compute partial derivatives of the function \( w = x \sin(yz^2) \), we first look at the effect each variable has on \( w \) while keeping the others constant.

Here are the three partial derivatives computed in this exercise:

• \( \frac{\partial w}{\partial x} \) measures how \(w\) changes with changes in \(x\) while keeping \(y \) and \(z\) constant. It's given as \( \sin(yz^2) \).
• \( \frac{\partial w}{\partial y} \) looks at change in \(w\) with respect to \(y\), producing \( xz^2 \cos(yz^2) \).
• \( \frac{\partial w}{\partial z} \) considers changes in \(w\) due to \(z\), leading to \( 2xyz \cos(yz^2) \).

These partial derivatives are fundamental in obtaining the overall rate of change of \(w\) using the chain rule.

Differentiation is a vital calculus technique used to calculate rates of change and slopes of curves. In the context of the chain rule, it becomes a multi-step process to find the total derivative. The chain rule is crucial due to its ability to connect these partial derivatives to produce a total derivative with respect to a particular variable—in this case, \( t \).Let’s breakdown the differentiation technique applied:

• Begin by defining \(w = \cos t \sin(t^2 e^{2t})\) and differentiating it using chain and product rules.
• The chain rule here is used to link the rate of change of \( w \) with \(x, y, z\) and combines them elegantly into a single expression in terms of \( t \).

By differentiating \(w\) as a function of \(x, y, z\), we can substitute back the expressions for \(x, y, z\), then calculate the derivatives \( \frac{dx}{dt} = -\sin t \), \( \frac{dy}{dt} = 2t \), and \( \frac{dz}{dt} = e^t\).
This solidifies our connection between partial derivatives and the total derivative via the chain rule.This process highlights how powerful differentiation techniques aid in solving complex problems beyond simple one-variable functions.

The rate of change is a crucial concept in calculus that describes how a quantity changes with respect to another. For functions of several variables, like \(w = x \sin(y z^2)\), the chain rule helps find the overall rate of change in a specific variable, here \( t \).To find this rate:

• We used the partial derivatives with respect to each variable and their rates of change concerning \(t\).
• Then, these rates are compounded into one by the chain rule to provide \( \frac{dw}{dt} \), the total rate of change in \(w\) as \(t\) varies.

The substitution of \( t = 0 \) into our derived expression shows how the function's rate of change behaves at this particular point. Setting \(t = 0\) simplifies the problem, leveraging known values like \(\cos 0 = 1\) and \(e^0 = 1\), leading to a zero rate of change here.Understanding the rate of change helps us predict the behavior of the function \( w \) and verify the correctness of our differentiated expressions by re-evaluating \(w(t)\) directly and proving consistency across methods.