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Chapter 1: Problem 18

Differentiate. $$y=(x-1)^{3}+(x+2)^{4}$$

Short Answer

Expert verified
\( y' = 3(x-1)^2 + 4(x+2)^3 \)

Step by step solution

01

Understand the Problem

This problem requires differentiating the function \( y = (x-1)^3 + (x+2)^4 \) with respect to \( x \).
02

Apply the Sum Rule

The Sum Rule states that the derivative of a sum is the sum of the derivatives. Thus, differentiate the two terms separately: \( f(x) = (x-1)^3 \) and \( g(x) = (x+2)^4 \) can be handled individually.
03

Differentiate the First Term

Use the Chain Rule to differentiate \( f(x) = (x-1)^3 \). Let \( u = x-1 \). Thus, \( f(x) = u^3 \). The derivative \( f'(x) = 3u^2 \times \frac{du}{dx} \). Since \( \frac{du}{dx} = 1 \), \( f'(x) = 3(x-1)^2 \).
04

Differentiate the Second Term

Similarly, use the Chain Rule to differentiate \( g(x) = (x+2)^4 \). Let \( v = x+2 \). Thus, \( g(x) = v^4 \). The derivative \( g'(x) = 4v^3 \times \frac{dv}{dx} \). Since \( \frac{dv}{dx} = 1 \), \( g'(x) = 4(x+2)^3 \).
05

Combine the Derivatives

Now combine the derivatives of the two terms: \( y' = f'(x) + g'(x) = 3(x-1)^2 + 4(x+2)^3 \).

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Key Concepts

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

Sum Rule in Differentiation
When you're dealing with the differentiation of a sum of functions, the Sum Rule comes to the rescue. The Sum Rule states that the derivative of a sum is simply the sum of the derivatives of each function.

For instance, if you have a function like \(y = f(x) + g(x)\), you can differentiate it term-by-term:
\[ y' = f'(x) + g'(x) \]
This makes differentiation much easier and more manageable. So, in the exercise at hand, where we need to differentiate \(y = (x-1)^3 + (x+2)^4\), the Sum Rule tells us to take the derivative of \((x-1)^3\) separately and the derivative of \((x+2)^4\) separately. Later on, we sum these individual results to get the overall derivative.
Chain Rule
The Chain Rule is a powerful tool in differentiation, especially when dealing with composite functions. It helps you find the derivative of a function nested within another function.

Let's break it down:
  • Suppose you have a composite function \(y = h(g(x))\).
  • The Chain Rule tells us to differentiate the inner function \(g(x)\) first, and then differentiate the outer function \(h(u)\) with respect to \(u\).
  • Combine these derivatives by multiplying them. So, \[ y' = h'(g(x)) \times g'(x) \]
In our exercise, we use the Chain Rule to differentiate each term. For \(f(x) = (x-1)^3\), we set \(u = x-1\), and for \(g(x) = (x+2)^4\), we set \(v = x+2\). Differentiating these inner functions \(u = x-1\) and \(v = x+2\) gives us \(\frac{du}{dx} = 1\) and \(\frac{dv}{dx} = 1\). Then, we find the derivatives of \(u^3\) and \(v^4\) with respect to \(u\) and \(v\), respectively, and multiply by the derivatives of \(u\) and \(v\).
Derivative of Polynomials
When dealing with polynomials, the differentiation process is straightforward, thanks to the power rule. The power rule states: for any term \(x^n\), its derivative is \(n \times x^{n-1}\).
  • In our example, for \(f(x) = (x-1)^3\), treat \(x-1\) as a single entity \(u\), making it \(u^3\).
  • Applying the power rule, the derivative becomes \(3u^2\), simplifying to \(3(x-1)^2\) since \(u = x-1\).
  • Similarly, for \(g(x) = (x+2)^4\), set \(v = x+2\), and the derivative becomes \(4v^3\), which is \(4(x+2)^3\) as \(v = x+2\).
The basics of polynomial differentiation make it simpler to find each term's derivative individually before combining them, fitting nicely with the Sum Rule.

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Most popular questions from this chapter

Let \(P(x)\) be the profit from producing (and selling) \(x\) units of goods. Match each question with the proper solution. Questions A. What is the profit from producing 1000 units of goods? B. At what level of production will the marginal profit be 1000 dollars? C. What is the marginal profit from producing 1000 units of goods? D. For what level of production will the profit be 1000 dollars? Solutions (a) Compute \(P^{\prime}(1000)\) (b) Find a value of \(a\) for which \(P^{\prime}(a)=1000\) (c) Set \(P(x)=1000\) and solve for \(x\) (d) Compute \(P(1000)\)

Compute the difference quotient $$\frac{f(x+h)-f(x)}{h}.$$ Simplify your answer as much as possible. $$f(x)=x^{2}-7$$

Let \(P(x)\) be the profit (in dollars) from manufacturing and selling \(x\) cars. Interpret \(P(100)=90,000\) and \(P^{\prime}(100)=1200 .\) Estimate the profit from manufacturing and selling 99 cars.

Using the sum rule and the constant-multiple rule, show that for any functions \(f(x)\) and \(g(x).\) $$\frac{d}{d x}[f(x)-g(x)]=\frac{d}{d x} f(x)-\frac{d}{d x} g(x).$$

Compute the following limits. $$\lim _{x \rightarrow-\infty} \frac{1}{x^{2}}$$
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