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Chapter 1: Problem 7

Find \(f^{\prime}(x)\) if \(f(x)=\left(\frac{x}{x+1}\right)^{4}\)

Short Answer

Expert verified
The derivative \(f^{\prime}(x)\) required by the exercise is \(4*(\frac{x}{x+1})^{3}*\frac{1}{(x+1)^{2}}\).

Step by step solution

01

Identify the Inside and Outside Functions

The main function can be written as \(f(x)=u^{4}\) where \(u=\frac{x}{x+1}\) is the inside function and \(u^{4}\) is the outside function.
02

Compute the Derivatives of the Inside and Outside Functions

The derivative of the outside function \(u^{4}\), according to the power rule, is \(4u^{3}\). For the inside function, the quotient rule is applied. For a function \(u=\frac{p}{q}\), its derivative is given by \(u'=\frac{p'q-q'p}{q^{2}}\). Deeming \(p=x\) and \(q=x+1\), we find the derivative of \(u\) is \(u'=\frac{(x+1)-x}{(x+1)^{2}}=\frac{1}{(x+1)^{2}}\).
03

Apply the Chain Rule

The chain rule states that \( (f(g(x)))' = f'(g(x))g'(x)\). Applying the derived expressions, we get \(f'(x) = 4u^{3}*\frac{1}{(x+1)^{2}} = 4*(\frac{x}{x+1})^{3}*\frac{1}{(x+1)^{2}}\).

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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 a fundamental tool in differentiation, heavily employed when dealing with functions of the form \( x^n \), where \( n \) is any real number. Derivatives are all about understanding how a function changes.
The power rule provides a simple formula: when you differentiate \( x^n \), you bring the exponent \( n \) down as a coefficient and subtract 1 from the exponent:
  • Derivative of \( x^n \) is \( nx^{n-1} \).
Imagine applying this rule to \( u^4 \), as was needed in the original exercise: you take the 4, multiply it by \( u^{4-1} \), giving you \( 4u^3 \).
Therefore, the power rule makes it a straightforward process to handle polynomial functions and numerous other expressions involving powers.
Quotient Rule
The quotient rule is essential when taking the derivative of a function that is the ratio of two differentiable functions, like \( \frac{p}{q} \).
When differentiating, the quotient rule states:
  • If \( u = \frac{p}{q} \), then \( u' = \frac{p'q - q'p}{q^2} \).
In the context of our exercise, where \( u = \frac{x}{x+1} \), the numerator \( p=x \) and denominator \( q=x+1 \) both must be differentiated.
It ensures that we correctly consider how each part of the fraction changes:
  • The derivative of \( p \) is 1 and of \( q \) is 1, leading to:
  • \( u' = \frac{(1)(x+1) - (1)(x)}{(x+1)^2} = \frac{1}{(x+1)^2} \).
This rule helps us get the proper rate of change for the entire fraction efficiently.
Chain Rule
The chain rule is a powerful technique to differentiate composite functions, functions wrapped around each other like nests of dolls.
When you have a function composed within another, say \( f(g(x)) \), the chain rule allows you to differentiate such complexities smoothly:
  • If \( (f \, o \, g)'(x) = f'(g(x))g'(x) \), with \( f \) as the outer function and \( g \) the inner function.
In the step-by-step solution, \( u \) was our inner function and \( u^4 \) the outer.
Applying the chain rule, you first find the derivative of the outer \( u^4 \) using the power rule to get \( 4u^3 \), and the inner \( u \) using the quotient rule to get \( \frac{1}{(x+1)^2} \).
Then these derivatives combine:
  • \( f'(x) = 4u^3 \times \frac{1}{(x+1)^2} = 4\left(\frac{x}{x+1}\right)^3 \times \frac{1}{(x+1)^2} \).
This interconnected application of rules speeds up navigating through calculus's layered structures.

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Most popular questions from this chapter

The average value of the function \(f(x)=(x-1)^{2}\) on the interval from \(x=1\) to \(x=5\) is (A) \(\frac{16}{3}\) (B) \(\frac{64}{3}\) (C) \(\frac{66}{3}\) (D) \(\frac{256}{3}\)

Find the volume of the solid that results when the region bounded by \(x= 1-y^{2}\) and the \(y\) -axis is revolved around the \(y\) -axis.

A radioactive element decays exponentially in proportion to its mass. One half of its original amount remains after \(5,750\) years. If \(10,000\) grams of the element are present initially, how much will be left after \(1,000\) years?

If the definite integral \(\int_{1}^{3}\left(x^{2}+1\right)\) is approximated by using the Trapezoid Rule with \(n=4,\) the error is (A) 0 (B) \(\frac{7}{3}\) (C) \(\frac{1}{12}\) (D) \(\frac{65}{6}\)

Evaluate the following integrals. \(\int \frac{\sec ^{2} x}{\tan x} d x\)
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