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Chapter 3: Problem 16

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

Short Answer

Expert verified
The derivative of the function \( f(x) = (4x+3)/(5x-2) ^{3} \) is \( 3(4x+3)^{2} * (5*(5x-2)*(4)-(4x+3)*5)/(5x-2)^{4} \).

Step by step solution

01

Identify the outer and inner functions

The outer function is \( (u/v)^{n} \) and the inner function is \( u/v \), where \( u = 4x + 3 \) and \( v = 5x - 2 \).
02

Apply the chain rule

By the chain rule, the derivative of \( f(x) \) is the derivative of the outer function times the derivative of the inner function. The derivative of the outer function with respect to \( u/v \) is \( 3(u/v)^{2} \). The derivative of the inner function \( u/v \) can be found using the quotient rule.
03

Apply the quotient rule

The derivative \( u'/v \) of \( u/v \) is \( (v*u'-u*v')/v^{2} \), according to the quotient rule. We already know that \( u = 4x + 3 \) and \( v = 5x - 2 \), so \( u' = 4 \) and \( v' = 5 \). Therefore, \( u'/v = (v*4 - (4x+3)*5)/(5x-2)^{2} \).
04

Compute the derivative of the function

Finally, the derivative of the function \( f(x) \) is the product of the derivative of the outer function and the derivative of the inner function, which is \( 3(u/v)^{2} * (v*4 - (4x+3)*5)/(5x-2)^{2} \). After substituting \( u = 4x + 3 \) and \( v = 5x - 2 \) into this expression and simplifying, we obtain the derivative of the function \( f(x) \).

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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 that deals with finding the rate at which a function is changing at any given point. It is the process of computing the derivative of a function. Intuitively, differentiation helps us understand how a small change in the input of a function affects the output.

For example, in physics, if we represent the position of an object as a function of time, differentiation can tell us the object's velocity at any moment鈥攅ssentially, how its position changes over time. Similarly, for the function in our exercise, differentiation allows us to find how the function's value changes as we vary the input variable, x.

The exercise introduces a complex function which involves a ratio raised to a power. This requires a combination of differentiation rules to solve. In particular, understanding the rules of differentiation, such as the chain rule and quotient rule, is essential to finding the derivative of such composite functions.
Quotient Rule
The quotient rule is a technique for differentiating functions that are fractions of two other functions. Given a function in the form of a quotient, \( \frac{u(x)}{v(x)} \), where both \( u \)

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Most popular questions from this chapter

Find equations for the lines normal to the hyperbola \(4 x^{2}-y^{2}=36\) that are parallel to the line \(2 x+5 y-4=0\).

Find \(f^{\prime \prime}(x)\). $$f(x)=\left(\frac{x}{1-x}\right)^{3}$$

Find a formula for the \(n\)th derivative. \(y=(a+b x)^{n} ; \quad n\) a positive integer, \(a, b\) constants.

Find the numbers \(x\) for which (a) \(f^{\prime \prime}(x)=0\), (b) \(f^{\prime \prime}(x)>0,\) (c) \(f^{\prime \prime}(x)<0\). $$f(x)=x^{3}$$

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