Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=5 x^{4 / 3}-\frac{2}{3} x^{3 / 2}+x^{2}-3 x+1\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 5x^{4/3} - \frac{2}{3}x^{3/2} + x^2 - 3x + 1\) is \(f'(x) = 20x^{\frac{1}{3}} - x^{\frac{1}{2}} + 2x - 3\).

Step 1: Identify terms to differentiate individually

First, break down the function \(f(x) = 5x^{4/3} - \frac{2}{3}x^{3/2} + x^2 - 3x + 1\) into individual terms that need to be differentiated. We have: 1. \(5x^{4/3}\) 2. \(-\frac{2}{3}x^{3/2}\) 3. \(x^2\) 4. \(-3x\) 5. \(1\)

Step 2: Apply the power rule

Apply the power rule to each term using the formula for differentiation of power functions: if \(g(x) = ax^n\), then \(g'(x) = nax^{n-1}\). 1. For \(5x^{4/3}\): \[\frac{d}{dx}\left(5x^{4/3}\right) = \frac{4}{3}(5)x^{\frac{4}{3} - 1} = 20x^{\frac{1}{3}}\] 2. For \(-\frac{2}{3}x^{3/2}\): \[\frac{d}{dx}\left(-\frac{2}{3}x^{3/2}\right) = \frac{3}{2}\left(-\frac{2}{3}\right)x^{\frac{3}{2} - 1} = -x^{\frac{1}{2}}\] 3. For \(x^2\): \[\frac{d}{dx}(x^2) = 2x^{2-1} = 2x\] 4. For \(-3x\): \[\frac{d}{dx}(-3x) = -3x^{1-1} = -3\] 5. For \(1\): \[\frac{d}{dx}(1) = 0\]

Step 3: Combine the derivatives

Combine the derivatives of each term found in step 2 using the sum/difference rule to find the derivative of the entire function: \[f'(x) = 20x^{\frac{1}{3}} - x^{\frac{1}{2}} + 2x - 3\]

Key Concepts

Functions

Functions are the building blocks of calculus and represent relationships between two sets of numbers or variables. A function is defined as a rule that assigns each input exactly one output. The input is often represented by the variable \(x\), and the output by \(f(x)\).
For example, in our original exercise, we have a function \(f(x)=5 x^{4 / 3}-\frac{2}{3} x^{3 / 2}+x^{2}-3 x+1\). This expression defines a rule that relates each input \(x\) to a specific output \(f(x)\), calculated by applying this formula. Functions are versatile and can model various real-world scenarios, from simple linear relationships to complex polynomial expressions.
Understanding the different types of functions and how they are constructed allows us to interpret and solve numerous problems. In calculus, functions are crucial because we often need to find rates of change, which requires calculating derivatives.

Differentiation

Differentiation is a key process in calculus, where we calculate the derivative of a function. The derivative provides us with the rate at which the function's value changes concerning its input variable, typically \(x\).
To differentiate a function, we apply rules that simplify finding the derivative, like the power rule or the product rule.
Using differentiation, we can determine several important aspects of a function:

• The slope or gradient of a curve at any point.
• Where the function is increasing or decreasing.
• Identify maxima, minima, and points of inflection.

In the original exercise, differentiation is used to find the derivative \(f'(x)\) of a given function \(f(x)\). Step-by-step, each term is differentiated individually by applying rules which make this complex-seeming operation become straightforward.

Power Rule

The power rule is a fundamental differentiation rule used extensively in calculus to find the derivative of functions that are polynomials or have powers. It states that, for a term of the form \(ax^n\), the derivative is given by \(nax^{n-1}\).
This is highly useful because it simplifies the differentiation process involving terms with power functions. For instance, in our given function \(f(x)\), we have several terms like \(5x^{4/3}\) and \(x^2\).
Applying the power rule allows us to derive these quickly:

• \(\frac{d}{dx}(5x^{4/3}) = 20x^{1/3}\)
• \(\frac{d}{dx}(x^2) = 2x\)

Understanding the power rule is crucial because many functions we encounter involve power terms. Mastery of this rule greatly aids in handling more complex differentiations that may involve additional rules, such as the product or chain rules.