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Chapter 2: Problem 19

Use the Generalized Power Rule to find the derivative of each function. $$w(z)=\sqrt[3]{9 z-1}$$

Short Answer

Expert verified
The derivative is \( w'(z) = \frac{3}{(9z-1)^{2/3}} \).

Step by step solution

01

Rewrite the function with exponents

The given function is \( w(z)=\sqrt[3]{9z-1} \). First, rewrite this radical expression by using exponents: \( w(z)=(9z-1)^{1/3} \).
02

Apply the Generalized Power Rule

Recall the Generalized Power Rule for derivatives: If \( f(z) = [g(z)]^n \), then the derivative \( f'(z) \) is \( n[g(z)]^{n-1}g'(z) \). Identify \( n=\frac{1}{3} \) and \( g(z)=9z-1 \).
03

Differentiate the inside function

Calculate the derivative of the inside function \( g(z)=9z-1 \). The derivative \( g'(z) \) is \( 9 \).
04

Apply the Generalized Power Rule

Using the results from Steps 2 and 3, apply the Generalized Power Rule:\[ w'(z) = \frac{1}{3}(9z-1)^{-2/3} \times 9 = 3(9z-1)^{-2/3} \].
05

Simplify the expression

Simplify the expression for the derivative:\[ w'(z) = \frac{3}{(9z-1)^{2/3}} \].

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Key Concepts

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

Derivative
In calculus, a derivative represents how a function changes as its input changes. It's like measuring the speed of a car, telling us how fast the position of the car changes with time. When given a function, such as the one in the exercise, the task of finding its derivative is asking us to determine the rate of change of that function with respect to its variable.
  • Rate of Change: The derivative provides insight into how a quantity varies with another. In mathematical terms, it's often noted as \( f'(x) \) or \( \frac{df}{dx} \).
  • Slope: For functions graphed as curves, the derivative at any point corresponds to the slope of the tangent line at that point.
Understanding derivatives is crucial for solving many mathematical problems, especially those involving optimization and motion. In our specific exercise, we've applied these concepts to a function with a fractional exponent, showcasing the importance of the derivative in a broader calculus context.
Exponentiation
Exponentiation involves raising a number to the power of another. In other words, it tells you how many times to use a number in a multiplication. When dealing with calculus, particularly with the Generalized Power Rule, it's common to encounter functions expressed exponentially.
  • Fractional Exponents: In this exercise, the exponent \( \frac{1}{3} \) represents the cube root of the inside function \( 9z-1 \). Rewriting roots as fractional exponents simplifies the differentiation process.
  • Rules of Exponents: Products, powers, and roots follow specific mathematical rules that help transform expressions. Knowing these rules is essential for simplifying complex derivatives.
By converting roots into fractional exponents, the Generalized Power Rule becomes easier to apply, allowing us to efficiently compute the derivative in a streamlined manner.
Calculus
Calculus is a broad field of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It serves as a powerful tool to model and analyze change over time and space, forming the foundation for much of modern science and engineering.
  • Differentiation: A core concept where we compute the derivative of functions, as we did using the Generalized Power Rule in our exercise.
  • Generalized Power Rule: A crucial derivative rule used when differentiating functions raised to a power. This enables us to deal with a variety of exponent forms, including roots and fractional powers.
Understanding these principles within calculus allows us to tackle a wide range of practical and theoretical problems. From physics to economics, calculus can describe and predict behaviors by analyzing the rates of change, showing its versatility and profound impact.

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Most popular questions from this chapter

Find an expression for the derivative of the composition of three functions, \(\frac{d}{d x} f(g(h(x)))\). [Hint: Use the Chain Rule twice.]

True or False: \(\frac{d}{d x} f(x / 2)=\frac{f^{\prime}(x / 2)}{2}\).

Find functions \(f\) and \(g\) such that the given function is the composition \(f(g(x))\). $$\left(x^{2}-x\right)^{-3}$$

A rocket can rise to a height of \(h(t)=t^{3}+0.5 t^{2}\) feet in \(t\) seconds. Find its velocity and acceleration 10 seconds after it is launched.

Use the Generalized Power Rule to find the derivative of each function. $$y=\left(\frac{1}{w^{4}+1}\right)^{5}$$
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