Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Rewrite each of these functions in a form that allows you to differentiate using the tools you have now, but with as little exertion as possible. Then differentiate. Work on strategy; none of these problems require the Quotient Rule. (a) \(f(x)=\frac{\frac{1}{x}+1}{\frac{x+1}{x^{2}+2 x}}\) (b) \(f(x)=\frac{\left(x^{3}+3 x\right)}{2 x^{4}}\)

Step-by-Step Solution

Answer

(a) The derivative of \(f(x)=\frac{\frac{1}{x}+1}{\frac{x+1}{x^{2}+2 x}}\) is \(f'(x) = 2x + 3\). (b) The derivative of \(f(x)=\frac{\left(x^{3}+3 x\right)}{2 x^{4}}\) is \(f'(x)=-\frac{1}{2}x^{-2} - \frac{9}{2}x^{-4}\).

Step 1: Problem (a) Step 1: Simplify the Function

Simplify the given function by using the property of division of fractions, where \(\frac{a}{b} = a \cdot \frac{1}{b}\). We apply this rule twice because the function \(f(x)\) is composed of two fractions. Hence, the function becomes:\(f(x) = \left(\frac{1}{x} + 1\right) \cdot \left(\frac{x^{2} + 2x}{x + 1}\right)\) Then, apply the distributive property to further simplify:\(f(x)=(x + 2) + (x^{2} + 2x)\)

Step 2: Problem (a) Step 2: Differentiate the Simplified Function

Now the simplified function can be differentiated using basic rules of differentiation. For the term \(x\), the derivative is 1. For \(2\), being a constant, the derivative is 0. The term \(x^{2}\) is differentiated using the Power Rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\), yielding 2x. For the term \(2x\), the derivative is \(2\). Adding these up, the derivative of the function, denoted as \(f'(x)\), becomes:\(f'(x)= 1 + 0 + 2x + 2 = 2x + 3\)

Step 3: Problem (b) Step 1: Simplify the Function

The next function \(f(x)=\frac{\left(x^{3}+3 x\right)}{2 x^{4}}\) is simplified by dividing each term in the numerator by the denominator:\(f(x)=\frac{1}{2}x^{-1} + \frac{3}{2}x^{-3}\)

Step 4: Problem (b) Step 2: Differentiate the Simplified Function

The simplified function can now be differentiated. For the term \(x^{-1}\), the Power Rule gives \(-x^{-2}\) and for the term \(x^{-3}\), the power rule gives \(-3x^{-4}\). Multiplying the result of differentiation with the coefficients in the original simplified function, it becomes:\(f'(x)=-\frac{1}{2}x^{-2} - \frac{9}{2}x^{-4}\)

Key Concepts

Simplifying Functions

When faced with complex functions that need differentiation, simplifying them first can make the entire process much more manageable. Simplification involves rewriting the function in a form that is easier to work with using basic algebraic rules and properties. Consider splitting fractions or breaking down terms whenever possible.

For example, let's look at the function given in Problem (a):

• The original function is a fraction of fractions: \( f(x) = \frac{\frac{1}{x} + 1}{\frac{x+1}{x^2+2x}} \).
• By applying the property \( \frac{a}{b} = a \cdot \frac{1}{b} \), you can rewrite the function as a multiplication problem: \( (\frac{1}{x} + 1) \cdot (\frac{x^2 + 2x}{x + 1}) \).
• Subsequently, use the distributive property to simplify further, resulting in \( f(x) = (x + 2) + (x^2 + 2x) \).

This new representation is much easier to differentiate since each component follows basic mathematical operations.

Power Rule

The Power Rule is a cornerstone of differentiation, crucial for understanding how to find derivatives of polynomial functions. Simply put, the Power Rule states that if you have a function \( f(x) = x^n \), its derivative is given by \( f'(x) = nx^{n-1} \). This rule simplifies the task of differentiation and is widely applicable.

In Problem (a), after simplifying, the Power Rule helps us find the derivatives of parts of the function:

• For \( x^2 \), using the Power Rule, the derivative is \( 2x \).
• If there's a term like \( x \), its derivative is straightforwardly \( 1 \), as \( x^1 \) induces a reduction to the constant multiplier.

The Power Rule eliminates the complexity seen in other differential calculus techniques and is particularly beneficial when dealing with polynomials.

Derivatives of Fractions

When differentiating functions that appear as fractions, it's often helpful to initially simplify them using algebraic techniques, making them more suitable for differentiation. Once simplified, these fractions often become straightforward polynomial functions.

Let's take a look at Problem (b):

• The function \( f(x) = \frac{x^3 + 3x}{2x^4} \) initially is a fraction.
• By dividing each term of the numerator by the denominator, you get: \( \frac{1}{2}x^{-1} + \frac{3}{2}x^{-3} \).
• This step transforms the fraction into a form where differentiation via the Power Rule is easily applicable.
• For \( x^{-1} \), the Power Rule yields \(-x^{-2} \), and for \( x^{-3} \), it gives \(-3x^{-4} \).
• Finally, apply the coefficients and construct the derivative \( f'(x) = -\frac{1}{2}x^{-2} - \frac{9}{2}x^{-4} \).

Thus, simplifying a fraction before differentiating simplifies the problem and makes the application of rules like the Power Rule more straightforward.