Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f^{\prime}(x) .\) Strategize to minimize your work. For example, \(\frac{x^{2}+3}{3 x}\) does not require the Quotient Rule. \(\frac{x^{2}+3}{3 x}=\frac{x}{3}+\frac{1}{x}=\frac{1}{3} x+x^{-1} .\) This is simpler to differentiate. $$ f(x)=\pi\left(3 x^{2}+7 x+1\right)(x-2) $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\pi(3x^{2}+7x+1)(x-2)\) is \(f^{\prime}(x)=\pi(9x^{2}+2x-13)\)

Step 1: Expand the Function

To simplify the function, let's expand the given function, \(f(x)=\pi(3x^{2}+7x+1)(x-2)\), first. Using the distributive property, the expanded form becomes \(f(x)=\pi(3x^{3}-6x^{2}+7x^{2}-14x+x-2)=\pi(3x^{3}+x^{2}-13x-2)\)

Step 2: Differentiate the Function

Next, differentiate this expanded function \(\pi(3x^{3}+x^{2}-13x-2)\) using the Power Rule. The derivative would be \(f^{\prime}(x)=\pi(9x^{2}+2x-13)\)

Step 3: Simplify the Derivative

Finally, simplify the derivative. Since \(\pi\) is a constant, it can be pulled out in front of the derivative. Hence, the final derivative simplifies to \(f^{\prime}(x)=\pi(9x^{2}+2x-13)\)

Key Concepts

Power Rule

The Power Rule is an essential tool in calculus for finding derivatives of functions where variables have exponent terms. It states that if you have a function of the form \( f(x) = x^n \), the derivative \( f'(x) \) is \( nx^{n-1} \). This rule simplifies the differentiation process immensely by providing a straightforward formula.
If your function includes several terms like \( 3x^3, x^2, \) or \( -13x \), apply the Power Rule to each term separately. For example, in \( 3x^3 \), the derivative is \( 9x^2 \) because you multiply the exponent 3 by the coefficient 3 and then decrease the exponent by one.
The Power Rule enables us to quickly handle terms in polynomials, making our calculations much less tedious. Remember that constants, like the \( -2 \) in the function, have a derivative of zero as they do not change with \( x \). This simplicity is why the Power Rule is a first step in differentiating polynomials.

Quotient Rule

The Quotient Rule is used in differentiation when dealing with the division of two functions. However, in some cases, like in the provided exercise, using the Quotient Rule might not be necessary. Instead, simplify the expression first, usually recommended when it leads to a simpler form.
The Quotient Rule states that for a function \( \frac{u(x)}{v(x)} \), the derivative \( f'(x) \) is \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). It's a useful rule, but as seen in the exercise, sometimes simplifying the function before differentiation can save work and reduce error potential.
For instance, in the example \( \frac{x^2+3}{3x} = \frac{x}{3} + \frac{1}{x} \), simply breaking down the expression made the differentiation process more manageable. Recognizing when to use such simplifications over direct application of the Quotient Rule is part of developing skill in calculus.

distributive property

The distributive property is a foundational principle in algebra that allows us to multiply a single term across terms inside a parenthesis. In calculus, it often prepares a function for differentiation.
Applied in expanding functions, the distributive property helps to simplify expressions into a form ready for the Power Rule. For instance, expanding \( \pi(3x^{2}+7x+1)(x-2) \) into \( \pi(3x^3 + x^2 - 13x - 2) \) allows direct differentiation.
Here's how it works:

• Multiply each term inside the parenthesis by each term in the other parenthesis.
• Combine like terms to simplify the expression.

With the simplified expression, you can then readily apply the Power Rule. The distributive property reduces complexity, helps in breaking down harder problems, and is indispensable for more manageable calculus work.

function expansion

Function expansion is rewriting a complex function as a polynomial or a sum of terms, which is often easier to handle, especially for differentiation. It involves applying algebraic skills like the distributive property.
In the original exercise, expanding \( \pi(3x^{2}+7x+1)(x-2) \) into \( \pi(3x^3 + x^2 - 13x - 2) \) illustrates function expansion. Once expanded, you can use simple rules like the Power Rule to find derivatives.
The process entails:

• Breaking down a product of terms using the distributive property.
• Combining and simplifying the resulting terms to get a polynomial.

This step prepares functions for methods like the Power Rule, enabling straightforward differentiation of each term. Mastering function expansion allows for tackling various mathematics problems with greater ease and confidence.