Problem 7 - 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. $$ f(x)=\left(\frac{5 x^{2}}{2}+7 x^{5}-5 x\right) x $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=(\frac{5 x^{2}}{2}+7 x^{5}-5 x) x\) is \(f^{\prime} (x) = \frac{15}{2} x^{2} + 42 x^{5} - 10x\).

Step 1: Simplify the Function

Begin by simplifying the given function. This can be done by multiplying out the expressions in \( f(x) = (\frac{5 x^{2}}{2}+7 x^{5}-5 x) x \). Here are the results of the multiplication: \n \( = \frac{5 x^{3}}{2}+ 7 x^{6} - 5x^{2} \)

Step 2: Apply the Power Rule

The Power Rule states that the derivative of \( x^n \), where n is any real number, is \( nx^{n-1} \). Apply this rule to differentiate each term in the simplified function from step 1. Differentiate the first term: \( \frac{d}{dx} (\frac{5 x^{3}}{2}) = \frac{3}{2} * 5x^{3-1} = \frac{15}{2}x^{2} \), the second term: \( \frac{d}{dx} (7x^{6}) = 6 * 7x^{6-1} = 42x^{5} \), and the third term: \( \frac{d}{dx} (-5x^{2} ) = 2 * -5x^{2-1} = -10x \)

Step 3: Write the Result

Combine the results from step 2 to obtain the final result. When combined, you should get the derivative \( f^{\prime} (x) = \frac{15}{2} x^{2} + 42 x^{5} - 10x \)

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus, crucial for finding derivatives of polynomial functions. The rule states that if you're differentiating a term of the form \(x^n\), where \(n\) can be any real number, the derivative is \(nx^{n-1}\). Simply put, you bring the exponent down in front of the term and subtract one from the exponent.
For example, when you have \(x^3\), applying the power rule gives you \(3x^{3-1} = 3x^2\). This rule simplifies the process of differentiation significantly, as it allows you to handle each term of a polynomial independently.
When given the function \( f(x)= \frac{5 x^{3}}{2}+ 7 x^{6} - 5x^{2} \), each term is a straightforward application of the power rule:

• The first term: \(\frac{d}{dx}(\frac{5x^3}{2}) = \frac{3}{2} \times 5x^{3-1} = \frac{15}{2}x^2\)
• The second term: \(\frac{d}{dx}(7x^6) = 6 \times 7x^{6-1} = 42x^5\)
• The third term: \(\frac{d}{dx}(-5x^2) = 2 \times -5x^{2-1} = -10x\)

Mastering the power rule is essential as it is one of the most frequently used tools in calculus.

Derivative

A derivative represents the rate at which a function is changing at any given point, essentially capturing the concept of instantaneous change. In simpler terms, it tells us how a function's output value changes as its input value changes.
Derivatives are fundamental in various fields such as mathematics, physics, and engineering because they allow us to find slopes of tangent lines, velocities, and rates of change.
Calculating a derivative involves finding a formula that describes this rate of change for any point in the function. In the case of the function \(f(x) = (\frac{5 x^{2}}{2}+7 x^{5}-5 x) x\), the derivative \(f'(x)\) is determined after the function is simplified and differentiated term by term.
With the function simplified to \(\frac{5 x^{3}}{2} + 7 x^{6} - 5x^{2}\), each term is differentiated using the power rule, resulting in the derivative \(f^{\prime} (x) = \frac{15}{2} x^{2} + 42 x^{5} - 10x\). This derivative gives the slope of the tangent line to the curve at any point \(x\).

Simplifying Functions

Simplifying functions is an essential step before differentiation as it can often reveal a more manageable form of the function, making the process easier. When a function is simplified, it reduces complexity and helps in applying differentiation rules more effectively.
In the given exercise, the function \(f(x) = (\frac{5 x^{2}}{2}+7 x^{5}-5 x) x\) was simplified by multiplying through by \(x\), resulting in \(\frac{5 x^{3}}{2} + 7 x^{6} - 5x^{2}\). This simplified expression allows direct application of the power rule without needing to apply the quotient or product rules.
Simplification can involve various techniques such as:

• Expanding polynomial expressions by distributing terms.
• Combining like terms to reduce the number of terms.
• Rewriting fractions to simpler forms when possible.

By organizing a function into a more straightforward form, you can make differentiation less prone to errors and more time-efficient, especially for students new to calculus.