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Chapter 3: Problem 26

Find the derivative. \(k(z)=\frac{6}{z^{2}+z-1}\)

Short Answer

Expert verified
The derivative is \( \frac{-12z - 6}{(z^2 + z - 1)^2} \).

Step by step solution

01

Identify the Function Type

The function is a rational function because it involves a polynomial in the numerator and another in the denominator. Specifically, it is of the form \( \frac{a}{g(z)} \) where \( g(z) = z^2 + z - 1 \).
02

Recall the Quotient Rule

To differentiate a quotient \( \frac{f(z)}{g(z)} \), use the quotient rule: \( \left(\frac{f(z)}{g(z)}\right)' = \frac{f'(z)g(z) - f(z)g'(z)}{g(z)^2} \), where \( f(z) = 6 \) and \( g(z) = z^2 + z - 1 \).
03

Differentiate the Numerator and Denominator

Differentiate the numerator \( f(z) = 6 \). Since 6 is a constant, its derivative is 0, \( f'(z) = 0 \). Differentiate the denominator \( g(z) = z^2 + z - 1 \) using the sum rule. Its derivative is \( 2z + 1 \).
04

Apply the Quotient Rule

Substitute the derivatives into the quotient rule formula: \( \left(\frac{6}{z^2 + z - 1}\right)' = \frac{0 \cdot (z^2 + z - 1) - 6 \cdot (2z + 1)}{(z^2 + z - 1)^2} \). Simplify the expression to get \( \frac{-6(2z + 1)}{(z^2 + z - 1)^2} \).
05

Simplify the Expression

Distribute the \( -6 \) in the numerator: \( -6(2z + 1) = -12z - 6 \). Therefore, the derivative is \( \frac{-12z - 6}{(z^2 + z - 1)^2} \).

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Key Concepts

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

Quotient Rule
The Quotient Rule is a formula used in calculus to find the derivative of a division of two functions. If you have a function in the form \(\frac{f(z)}{g(z)}\), the derivative is calculated as \(\left(\frac{f(z)}{g(z)}\right)' = \frac{f'(z)g(z) - f(z)g'(z)}{g(z)^2}\).
  • \(f'(z)\) and \(g'(z)\) are the derivatives of the numerator and the denominator, respectively.
  • The formula ensures that each component is differentiated separately and then combined.
To apply the quotient rule, differentiate each part individually, use the formula, then simplify the resulting expression. This is particularly useful when dealing with rational functions, where both the numerator and denominator are polynomials.
Rational Functions
Rational functions are fractions where both the numerator and the denominator are polynomials. An example of a rational function is \(k(z) = \frac{6}{z^2 + z - 1}\) as it includes a constant numerator (which can be considered a polynomial of degree 0) and a polynomial denominator.
  • They can exhibit a variety of behaviors; they can spike or dip at certain points, or asymptotically approach values as the input grows large.
  • The complexity of rational functions often makes them interesting but challenging to differentiate.
Despite their complexity, differentiating them becomes manageable with the quotient rule, which directly applies to any situation where polynomials are divided.
Polynomial Derivative
When differentiating polynomials, you apply the power rule to each term individually. The derivative of a term like \(az^n\) is found using \(naz^{n-1}\). This means for each term of the polynomial:
  • Increase in powers of \(z\) indicates the number of times the chain rule is applied.
  • Coefficients are reduced in line with their variable's power reduction.
For instance, with \(g(z) = z^2 + z - 1\), its derivative, \(g'(z)\), becomes \(2z + 1\), using these rules. Differentiation of polynomials forms the basis for applying the quotient rule to rational functions, as seen in this exercise.

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Most popular questions from this chapter

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