/*! 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 40 Find \(f^{\prime \prime}(x)\). ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 40

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

Short Answer

Expert verified
The second derivative of the function \(f(x) = (x^2 - 5x + 2)^2\) is \(f^{\prime \prime}(x) = 6x^2 - 30x + 20\).

Step by step solution

01

Compute the first derivative

The first derivative of \(f(x)\) can be found using the chain rule, which is \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \). So, \(g(x) = x^2 - 5x + 2\) and \(f(u) = u^2\), then \(f'(u) = 2u\). Applying the chain rule, \(f'(x) = 2(x^2 - 5x + 2)(2x - 5)\).
02

Simplify the first derivative

Now, simplify the expression by expanding the products to get \(f'(x) = 2(2x^3 - 15x^2 + 20x -10)\).
03

Compute the second derivative

Take derivative again to find second derivative, which is \(f^{\prime \prime}(x) = 6x^2 - 30x + 20\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Chain Rule
The Chain Rule is a fundamental principle in calculus used to calculate the derivative of a composite function. When a function is composed of one function inside of another, such as the case with nested functions, the Chain Rule allows us to differentiate it efficiently.

In our example, to find the first derivative of the function \(f(x)=(x^{2}-5x+2)^{2}\), we recognize that it is a composition of two functions: \(g(x) = x^2 - 5x + 2\) and \(f(u) = u^2\), where \(u = g(x)\). Applying the Chain Rule, we take the derivative of the outer function \(f(u)\) with respect to \(u\) to get \(f'(u) = 2u\), and then multiply by the derivative of the inner function \(g(x)\) with respect to \(x\) to get \(g'(x) = 2x - 5\). The Chain Rule formula \(f'(x) = f'(g(x)) \cdot g'(x)\) gives the derivative of the composite function.
Derivative Simplification
Once we have applied the Chain Rule to obtain the first derivative, we need to simplify the derivative to make it easier to handle and understand. Derivative simplification involves expanding products, combining like terms, and reducing complex expressions.

For instance, in our exercise after applying the Chain Rule, we ended up with \(f'(x) = 2(x^2 - 5x + 2)(2x - 5)\). Simplifying this yields \(f'(x) = 2(2x^3 - 10x^2 + 4x - 10x^2 + 25x -10)\) which can then be further simplified to \(f'(x) = 4x^3 - 30x^2 + 40x - 20\). It鈥檚 essential to perform this simplification to make the subsequent steps of finding the second derivative less complicated.
Calculus Problems
Calculus problems often involve finding derivatives and understanding the behavior of functions. These tasks can be challenging, but with practice, the key steps become more intuitive.

In this case, the process includes applying rules of differentiation (such as the Chain Rule), simplifying algebraic expressions, and understanding how to take second derivatives. After simplifying, we take the derivative of \(f'(x)\) to get \(f''(x)\).

Continually practicing these steps, and working through problems step by step, helps strengthen understanding and improve the ability to solve calculus problems more effectively. The goal is always to find accurate and simplified expressions that describe the rate at which quantities change, which is the heart of differential calculus.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find a function \(y=f(x)\) with the given derivative. Check your answer by differentiation. $$y^{\prime}=3\left(x^{2}+1\right)^{2}(2 x)$$

Find the points \((c, f(c))\) where the line tangent to the graph of \(f(x)=x^{3}-x\) is parallel to the secant line that passes through the points \((-1, f(-1))\) and \((2, f(2))\).

$$\text { Exercise } 77 \text { with } f(x)=\sin x-\sin ^{2} x \text { for } 0 \leq x \leq 2 \pi$$

The number \(a\) is called a triple zero (or a zero of multiplicity 3) of the polynomial \(P\) if $$P(x)=(x-a)^{3} q(x) \quad \text { and } \quad q(a) \neq 0$$ Prove that if \(a\) is a triple zero of \(P\), then \(a\) is a zero of \(P . P^{\prime}\) and \(P^{\prime \prime},\) and \(P^{\prime \prime \prime}(a) \neq 0\).

Use a CAS to express the following derivatives in \(f^{\prime}\) notation. (a) \(\frac{d}{d x}\left[f\left(\frac{1}{x}\right)\right]\) (b) \(\frac{d}{d x}\left[f\left(\frac{x^{2}-1}{x^{2}+1}\right)\right]\) (c) \(\frac{d}{d x}\left[\frac{f(x)}{1+f(x)}\right]\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.