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Chapter 2: Problem 26

Find the derivative of the algebraic function. $$ f(x)=x^{4}\left(1-\frac{2}{x+1}\right) $$

Short Answer

Expert verified
The derivative of the algebraic function \(f(x)=x^{4}\left(1-\frac{2}{x+1}\right)\) is \(f'(x) = x^{2}(4x-3)\)

Step by step solution

01

Simplify the function

Before jumping into deriving the function, it is crucial to simplify it first. simplify the function \(f(x)=x^{4}\left(1-\frac{2}{x+1}\right)\) to get it in a form that'll be easier to derive. Distribute the \(x^{4}\) term to the equation to get, \(f(x)=x^{4}-2x^{3}\)
02

Differentiate

Apply the Power Rule which states that the derivative of \(x^{n}\) is \(nx^{n-1}\). Using the power rule, differentiate \(f(x)=x^{4}-2x^{3}\) term-by-term to get the derivative of f, \(f'(x)= 4x^{3}-6x^{2}\)
03

Simplify the Derived Function

Finally, simplify the derivative function \(f'(x)= 4x^{3}-6x^{2}\), by taking out a common factor of \(x^2\), thus getting \(f'(x) = x^{2}(4x-3)\)

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Key Concepts

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

Algebraic Functions
When dealing with derivatives, we often encounter algebraic functions. These are composed of polynomials, rational functions, and combinations of these with "+" and "-" operations. An algebraic function like \( f(x) = x^4(1 - \frac{2}{x+1}) \) involves a polynomial and a rational expression. To differentiate efficiently, it鈥檚 beneficial to simplify the function first. Simplification can make the derivative process smoother, as seen in the example where we begin with this complex expression and simplify it to \( f(x) = x^4 - 2x^3 \). The easier the function looks, the simpler it is to apply derivative rules. Understanding how to manipulate algebraic expressions is key in calculus for simplifying and preparing them for differentiation.
Power Rule
The power rule is a fundamental concept in calculus that makes deriving polynomials straightforward. It states that the derivative of \( x^n \) is \( nx^{n-1} \). Knowing this rule helps you quickly find the slope of a given function. In our exercise, after simplifying the formula to \( f(x) = x^4 - 2x^3 \), we apply the power rule to find the derivative term-by-term:
  • For the term \( x^4 \), the derivative is \( 4x^3 \) because you multiply the power by the coefficient and then subtract one from the exponent.
  • For the term \( -2x^3 \), the derivative becomes \( -6x^2 \).
Using the power rule simplifies the laborious task of finding derivatives, especially in polynomial functions.
Simplification in Calculus
Simplification in calculus isn鈥檛 just about making the math tidier; it鈥檚 also about creating clarity for differentiation or integration. Once you derive a function, simplifying the resulting expression can provide a more straightforward solution, reveal common factors, or make further calculations easier.After deriving the function in our exercise, the result is \( f'(x) = 4x^3 - 6x^2 \). Even though this is already the derivative, simplifying it further by factoring out the common term \( x^2 \), gives us \( f'(x) = x^2(4x - 3) \). This step not only provides a cleaner form but shows underlying relationships between terms which can be useful in solving equations or finding roots. Simplification is an ongoing goal in calculus to ensure solutions are both elegant and practical.

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Most popular questions from this chapter

Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ g(x)=x^{2}+e^{x}, \quad[0,1] $$

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