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Magazine Chapter 2: Problem 23
Use the Generalized Power Rule to find the derivative of each function. $$y=\left(\frac{1}{w^{3}-1}\right)^{4}$$
Short Answer
Expert verified
The derivative is \(-12w^2 \frac{1}{(w^3 - 1)^5}\)."
Step by step solution
01
Identify the Function and the Generalized Power Rule
The function given is \( y = \left(\frac{1}{w^3 - 1}\right)^4 \). We define this as a composition of two functions: an outer function \( u^4 \) where \( u = \frac{1}{w^3 - 1} \). The Generalized Power Rule states that \( \frac{d}{dw} [u^n] = nu^{n-1}\frac{du}{dw} \).
02
Differentiate the Outer Function Using the Power Rule
First, apply the power rule to the outer function: \( \frac{d}{dw} [u^4] = 4u^3 \frac{du}{dw} \). Here, \( u = \frac{1}{w^3 - 1} \), so we keep the expression for \( \frac{du}{dw} \) in derivative form for now.
03
Differentiate the Inner Function Using the Quotient Rule
To differentiate \( u = \frac{1}{w^3 - 1} \), use the quotient rule, \( \frac{d}{dw} \left(\frac{1}{v}\right) = -\frac{v'}{v^2} \) where \( v = w^3 - 1 \) and \( v' = 3w^2 \). Hence, \( \frac{du}{dw} = -\frac{3w^2}{(w^3 - 1)^2} \).
04
Combine Both Derivatives
Substitute \( \frac{du}{dw} \) into the expression from Step 2: \( \frac{d}{dw} [u^4] = 4u^3 \left(-\frac{3w^2}{(w^3 - 1)^2}\right) \). Simplify to get \( -12w^2 \left(\frac{1}{w^3 - 1}\right)^3 \left(\frac{1}{(w^3 - 1)^2}\right) \).
05
Simplify the Expression
Combine the exponents: \( \frac{1}{(w^3 - 1)^3} \) from \( u^3 \) and \( \frac{1}{(w^3 - 1)^2} \), giving \( \frac{1}{(w^3 - 1)^5} \). Therefore, the derivative is \( -12w^2 \frac{1}{(w^3 - 1)^5} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Derivatives
Derivatives are a fundamental concept in calculus that measures how a function changes as its input changes. In simple terms, a derivative indicates a rate of change or how steep a curve is at any given point. This concept is essential in determining slopes of functions, which is why derivatives are hugely important in both mathematics and real-world applications like physics and engineering.
- Instantaneous Rate of Change: The derivative of a function at a specific point gives the instantaneous rate of change, similar to how speed indicates a car's velocity at a particular moment.
- Slope of Tangent: The derivative is also the slope of the tangent line to the curve of a function at a given point.
Applying the Quotient Rule
The quotient rule is a technique used to find the derivative of functions that are ratios of two other functions. This is crucial when dealing with fractions or rational functions in calculus and provides a systematic way to compute derivatives without manually expanding the function.
To apply the quotient rule, remember the formula: if you have a function in the form of \( \frac{f(x)}{g(x)} \), the derivative \( \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) \) is given by:
\[ \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} \]
To apply the quotient rule, remember the formula: if you have a function in the form of \( \frac{f(x)}{g(x)} \), the derivative \( \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) \) is given by:
\[ \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} \]
- Numerator and Denominator Derivatives: Calculate derivatives separately for the numerator and the denominator functions.
- Apply Formula: Substitute into the quotient rule formula, simplify as needed.
- Watch for Signs: Correctly apply negative signs and square the denominator to avoid calculation errors.
Importance of Calculus Education
Calculus is a foundational element of higher mathematics education and critical for students pursuing careers in science, technology, engineering, and mathematics (STEM) fields. Learning calculus helps develop problem-solving skills and abstract thinking, enabling students to understand and model complex systems.
- Analytical Skills: By resolving calculus problems, students enhance analytical capabilities, allowing them to dissect problems into manageable parts.
- Conceptual Understanding: Grasping calculus concepts such as limits, derivatives, and integrals provides a deeper insight into how mathematical functions and real-world phenomena interact.
- Applications: From predicting trends in economics to determining changes in biological systems, calculus is invaluable in a wide range of practical applications.
Exploring Mathematical Functions
Mathematical functions form the core around which most of calculus is built. A function is essentially a relation between a set of inputs and a set of possible outputs, with each input linked to one specific output. Functions allow us to model and analyze real-world situations mathematically.
- Types of Functions: Includes linear, quadratic, polynomial, rational, exponential, and logarithmic functions, each with its unique properties and applications.
- Composition of Functions: Involves combining two functions where the output of one function becomes the input of another, leading to complex functions that require advanced tools like the Generalized Power Rule for differentiation.
- Real-world Modeling: Functions help translate real-world scenarios into mathematical problems that can be solved, analyzed, and reported mathematically, such as modeling population growth or financial forecasts.
