/*! 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 31 Differentiate each function. $... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 31

Differentiate each function. $$f(x)=\left(\frac{3 x-1}{5 x+2}\right)^{4}$$

Short Answer

Expert verified
\[ f'(x) = \frac{44(3 x - 1)^{3}}{(5 x + 2)^{5}}. \]

Step by step solution

01

- Identify the function type

The function given is a composite function, specifically a power of a rational function.
02

- Apply the Chain Rule

We differentiate the function using the chain rule. If we denote the inner function as \[ u(x) = \frac{3 x - 1}{5 x + 2} \] and the outer function as \[ v(u) = u^4, \] then the differentiation process is: \[ \frac{d}{dx} f(x) = v'(u(x)) \times u'(x). \]
03

- Differentiate the outer function

Differentiate the outer function \( v(u) = u^4 \) with respect to \( u \): \[ v'(u) = 4u^3. \]
04

- Differentiate the inner function

Differentiate the inner function \[ u(x) = \frac{3 x - 1}{5 x + 2} \] using the quotient rule: \[ u'(x) = \frac{(5x+2)(3) - (3x-1)(5)}{(5x+2)^2}. \] Simplify the numerator: \[ \frac{(15x + 6) - (15x - 5)}{(5x+2)^2} = \frac{15x + 6 - 15x + 5}{(5x+2)^2} = \frac{11}{(5x+2)^2}. \]
05

- Combine the results

Substitute \(u\) and \(u'\) into the chain rule result: \[ \frac{d}{dx} f(x) = 4 \times \frac{3x - 1}{5x + 2}^3 \times \frac{11}{(5x + 2)^2}. \] Thus, the derivative is \[ f'(x) = \frac{44(3 x - 1)^{3}}{(5 x + 2)^{5}}. \]

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Key Concepts

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

Composite Functions
Composite functions involve two or more functions combined to form a new function. In this type of function, an 'inner function' is nested inside an 'outer function'. For instance, if we have a function g(x) and another function f(u), a composite function can be represented as f(g(x)). In our exercise, the function given is a composite function because it contains a rational function raised to a power. Understanding how to differentiate composite functions is essential, which is where the Chain Rule comes into play.
Quotient Rule
The Quotient Rule is used to differentiate functions that are the division of two other functions. If we have u(x) = \frac{A(x)}{B(x)}, then its derivative is found using the formula: \[ u'(x) = \frac{A'(x)B(x) - A(x)B'(x)}{[B(x)]^2} \]. In our step-by-step solution, the inner function u(x) = \frac{3x - 1}{5x + 2} is differentiated using the Quotient Rule. Here's how it works: First, identify the numerator as A(x) = 3x - 1 and its derivative A'(x) = 3. Next, identify the denominator as B(x) = 5x + 2 with its derivative B'(x) = 5.
Power Rule
The Power Rule is applied to differentiate functions where variables are raised to a power. For a function y = x^n, the derivative is given by \[ y' = n x^{n-1} \]. When applying this to our composite function, the outer function v(u) = u^4 is differentiated to get v'(u) = 4u^3. This is integral when using the Chain Rule. Combining the Power and Chain Rules allows us to differentiate more complex composite functions effectively. In our example, this results in v'(u) = 4\bigg(\frac{3x - 1}{5x + 2}\bigg)^3.

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Most popular questions from this chapter

Find the derivative of each of the following functions analytically. Then use a calculator to check the results. $$g(x)=\frac{4 x}{\sqrt{x-10}}$$

Differentiate each function. Let \(f(x)=\frac{x}{x+1}\) and \(g(x)=\frac{-1}{x+1}\) a) Compute \(f^{\prime}(x)\) b) Compute \(g^{\prime}(x)\) c) What can you conclude about \(f\) and \(g\) on the basis of your results from parts (a) and (b)?

Find \(d y / d x .\) Each function can be differentiated using the rules developed in this section, but some algebra may be required beforehand. $$y=(x-1)(x+1)$$

The following is the beginning of an alternative proof of the Quotient Rule that uses the Product Rule and the Power Rule. Complete the proof, giving reasons for each step. Proof. Let $$Q(x)=\frac{N(x)}{D(x)}$$ Then $$Q(x)=N(x) \cdot[D(x)]^{-1}$$ Therefore,

Use a graphing calculator to check the results of Exercises \(1-48\).
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