/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 228 Let \(z=x^{2} y, \quad\) where \... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 228

Let \(z=x^{2} y, \quad\) where \(x=t^{2}\) and \(y=t^{3} .\) Find \(\frac{d z}{d t}\)

Short Answer

Expert verified
The derivative \(\frac{d z}{d t} = 7t^{6}\).

Step by step solution

01

Express z in terms of t

Given that \(z = x^2 y\), where \(x = t^2\) and \(y = t^3\), substitute these expressions into \(z\) to get \(z = (t^2)^2 (t^3) = t^4 \, t^3 = t^7\).
02

Differentiate z with respect to t

Now that \(z = t^7\), find the derivative of \(z\) with respect to \(t\). Use the power rule for differentiation, which states that \(\frac{d}{dt}(t^n) = n t^{n-1}\). Therefore, \(\frac{d z}{d t} = 7t^{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is one of the most straightforward rules in differentiation. It helps you find the derivative of functions of the form \(f(x) = x^n\). Here鈥檚 how it works:
  • The power rule states that the derivative \(\frac{d}{dx}(x^n)\) is \(nx^{n-1}\).
  • This rule is particularly useful because it applies to any real number \(n\), whether it is a positive integer, negative integer, or even a fraction.
  • By simply reducing the power by one and multiplying by the original power, you quickly gain the derivative of a power function.
In our original problem, after expressing \(z\) in terms of \(t\) as \(t^7\), applying the power rule allows us to differentiate it easily. Thus, the derivative is \(7t^{6}\), showcasing the power rule鈥檚 efficiency in simplifying the process of finding derivatives for polynomial expressions.
Chain Rule
The chain rule is like a tool that allows you to find derivatives of composite functions. A composite function has one function nested inside another, like \(f(g(x))\). This rule is essential when you're dealing with functions that are composed of other functions.
  • The chain rule states that if you have a composite function \(f(g(x))\), its derivative is \((f'(g(x)))\cdot(g'(x))\).
  • In other words, you first find the derivative of the outer function using the inner function as the input, then multiply this by the derivative of the inner function.
  • This rule often shows up when variables are expressed in terms of other variables, which is common in physics and engineering problems.
For our exercise, though the intermediate step simplified \(z\) to \(t^7\), if we hadn't done this, we'd use the chain rule to derive derivatives by each nested function, like \(x\) and \(y\), all defined in terms of \(t\). It鈥檚 a powerful method to deal with more complex, layered differentiation problems.
Implicit Differentiation
Implicit differentiation is used when you have equations not solved for one variable neatly in terms of others. In simple cases, it can also be applied when it's hard to isolate a variable. Unlike explicit functions where one variable is explicitly expressed in terms of another, implicit functions involve multiple variables intertwined.
  • With implicit differentiation, you differentiate both sides of an equation with respect to a common variable.
  • While differentiating, treat all y's as functions of \(x\) (if you鈥檙e differentiating with respect to \(x\)) and apply the derivative rules you know, including the chain rule.
  • Whenever you take the derivative of \(y\), add \(\frac{dy}{dx}\) because \(y\) is a function of \(x\).
In the case of our problem, although implicit differentiation didn鈥檛 play a role, it's useful for dealing with related rates and equations that do not separate neatly into functions of one variable. Being comfortable with this concept enhances your problem-solving toolkit, especially for more complicated expressions.

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