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Chapter 1: Problem 34

Find \(y^{\prime \prime}\). $$y=\left(x^{3}-2\right)(5 x+1)$$

Short Answer

Expert verified
The second derivative is \(y'' = 60x^2 + 6x\).

Step by step solution

01

Expand the Expression

First expand the given function using the distributive property.\(y = (x^3 - 2)(5x + 1)\)Multiply each term in the first polynomial by each term in the second polynomial:\[y = x^3 \times 5x + x^3 \times 1 - 2 \times 5x - 2 \times 1\]Simplify each term:\[y = 5x^4 + x^3 - 10x - 2\]
02

Differentiate Once

Find the first derivative of the expanded function.\(y = 5x^4 + x^3 - 10x - 2\)Differentiate term-by-term:\[y' = \frac{d}{dx}(5x^4) + \frac{d}{dx}(x^3) - \frac{d}{dx}(10x) - \frac{d}{dx}(-2)\]Using basic differentiation rules:\[y' = 20x^3 + 3x^2 - 10\]
03

Differentiate Again

Find the second derivative of the function.\(y' = 20x^3 + 3x^2 - 10\)Differentiate term-by-term again:\[y'' = \frac{d}{dx}(20x^3) + \frac{d}{dx}(3x^2) - \frac{d}{dx}(-10)\]Using basic differentiation rules:\[y'' = 60x^2 + 6x\]

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Key Concepts

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

differentiation
Differentiation is a fundamental concept in calculus. It measures how a function changes as its input changes. Simply put, it finds the rate at which one quantity changes with respect to another. Here are some key points to remember:
  • The derivative of a function is represented as \(y'\) or \(\frac{dy}{dx}\).
  • It shows the slope of the function's graph at any point.
  • Basic differentiation rules include the power rule, product rule, quotient rule, and chain rule.
In this exercise, we utilize the power rule to differentiate polynomials. For example, if \(f(x) = x^n\), then \(f'(x) = nx^{n-1}\). We applied this to each term in the function to find both the first and second derivatives.
polynomial expansion
Polynomial expansion involves expressing a product of polynomials as a sum of terms. This is often necessary before differentiation. In our example, we expanded \(y = (x^3 - 2)(5x + 1)\) by multiplying each term in the first polynomial by each term in the second polynomial.
This process is broken down as follows:
  • Create products for each pair of terms: \(x^3 \times 5x\), \(x^3 \times 1\), \(-2 \times 5x\), \(-2 \times 1\).
  • Simplify each product: \(5x^4\), \(x^3\), \(-10x\), and \(-2\).
  • Combine these terms: \y = 5x^4 + x^3 - 10x - 2\.
Once expanded, the polynomial is ready for differentiation.
calculus
Calculus is the branch of mathematics that studies continuous change. It has two main branches: differential calculus and integral calculus. Differential calculus, which we applied here, focuses on the concept of a derivative. This gives us the rate of change and the slope of functions.
Let's break down our approach:
  • First, we expanded the polynomial to make it easier to differentiate.
  • Next, we applied differentiation rules to find the first derivative.
  • Finally, we differentiated again to find the second derivative.
All these steps illustrate fundamental calculus principles of finding and using derivatives to analyze the behavior of functions.

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Most popular questions from this chapter

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=(-4 x)^{3}$$

For each of the following, graph \(f\) and \(f^{\prime}\) and then determine \(f^{\prime}(1) .\) For Exercises use Deriv on the \(T I-83\). $$f(x)=x^{3}-2 x-2$$

Use the derivative to help show whether each function is always increasing, always decreasing, or neither. $$f(x)=x^{5}+x^{3}$$

Business and Economics A total-revenue function is given by \(R(x)=1000 \sqrt{x^{2}-0.1 x}\) where \(R(x)\) is the total revenue, in thousands of dollars, from the sale of \(x\) items. Find the rate at which total revenue is changing when 20 items have been sold.

First, use the Chain Rule to find the answer. Next, check your answer by finding \(f(g(x))\) taking the derivative, and substituting. \(f(u)=\frac{u+1}{u-1}, g(x)=u=\sqrt{x}\) Find \((f \circ g)^{\prime}(4)\)
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