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Chapter 3: Problem 18

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(z)=-\frac{1}{z^{6.1}}$$

Short Answer

Expert verified
The derivative is \( f'(z) = \frac{6.1}{z^{7.1}} \).

Step by step solution

01

Identify the Function's Form

The function given is in the form of a power function multiplied by a constant: \[ f(z) = -z^{-6.1} \]
02

Apply the Power Rule for Derivatives

The power rule for derivatives states that if \( f(z) = az^n \), then the derivative is \( f'(z) = anz^{n-1} \). Here, \( a = -1 \) and \( n = -6.1 \).
03

Differentiate the Function

Apply the power rule to differentiate the function:\[ f'(z) = -1 \cdot (-6.1) z^{-6.1 - 1} \] which simplifies to:\[ f'(z) = 6.1 z^{-7.1} \]
04

Simplify the Derivative

The final derivative is already simplified in terms of negative exponents. It can be expressed as:\[ f'(z) = \frac{6.1}{z^{7.1}} \] by rewriting the negative exponent as a positive exponent in the denominator.

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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 calculus, especially when working with polynomials or expressions involving exponents. It's a shortcut that helps you find the derivative of functions quickly.
Imagine you have a function of the form \( f(z) = az^n \). The power rule tells us the derivative \( f'(z) \) is \( anz^{n-1} \). Here's how it works:
  • "a" is the coefficient, which remains part of the derivative process.
  • "n" gets multiplied with "a" as you start the differentiation.
  • The exponent "n" gets reduced by one to form "n-1", which becomes the new exponent.
For example, using the power rule on \( f(z) = -z^{-6.1} \), we get the derivative \( f'(z) = 6.1 z^{-7.1} \), showcasing how easily the rule helps compute derivatives.
Negative Exponents
Negative exponents can often confuse, but they are just another way of expressing division. When you see a negative exponent such as \( z^{-n} \), it is equivalent to \( \frac{1}{z^n} \). This transformation is particularly useful when simplifying mathematical expressions.
Understanding negative exponents helps clarify the transition from a function's derivative in its raw form to one that's more intuitive or easier to read.
  • The original derivative \( f'(z) = 6.1 z^{-7.1} \) can be converted to \( \frac{6.1}{z^{7.1}} \).
  • This shift often simplifies integration or solving equations involving such terms.
The key is recognizing that switching between negative and positive exponents is a simple rearrangement.
Differentiation
Differentiation is the process of finding a derivative, which measures how a function changes as its input changes. Think of the derivative as the "rate of change" or the "slope of the curve" at any point.
Using differentiation, you can:
  • Calculate the speed of an object over time (velocity).
  • Determine the rate at which a population grows or declines.
  • Understand how a function behaves in general.
In the given problem, differentiation using the power rule allowed us to transform \( f(z) = -z^{-6.1} \) into its derivative \( f' (z) = 6.1 z^{-7.1} \), showing how the function's slope behaves as "z" changes.
Calculus
Calculus is the mathematical study of change, and it comprises two main branches: differentiation and integration. Understanding calculus is essential as it provides the tools to explore changes and motion in a precise manner.
Differentiation, as seen in this problem, allows you to examine the rate at which quantities change. This is useful across numerous fields:
  • Physics, for calculating forces and velocities.
  • Biology, to model population dynamics.
  • Economics, to assess changes in cost and revenue.
With calculus, you gain the ability to solve problems that involve variable rates of change, making it a vital component of advanced mathematics.

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