Problem 29 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{x^{2}+x^{3}}{x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 2x + 1 \).

Step 1: Simplify the Function

The first step is to simplify the function. The given function is \( f(x) = \frac{x^2 + x^3}{x} \). We can simplify this by dividing each term in the numerator by \( x \). \[ f(x) = \frac{x^3}{x} + \frac{x^2}{x} = x^2 + x \] This makes the function simpler for us to differentiate.

Step 2: Differentiate the Simplified Function

Next, we find the derivative of the simplified function \( f(x) = x^2 + x \). To do this, we apply the power rule: the derivative of \( x^n \) is \( nx^{n-1} \).- For \( x^2 \), the derivative is \( 2x^{2-1} = 2x \).- For \( x \), the derivative is \( 1 \).Hence, the derivative of \( f(x) = x^2 + x \) is: \[ f'(x) = 2x + 1 \]

Key Concepts

Simplify Functions

Before tackling the differentiation process, it's often helpful to simplify complicated functions. Simplifying means rewriting the function in its most basic form without changing its value. It makes the subsequent math operations easier.
For instance, consider the function \( f(x) = \frac{x^2 + x^3}{x} \). At first glance, this might look complex. But if you simplify, you rewrite as \( f(x) = x^2 + x \).
The process involves dividing each term in the numerator by the denominator:

• \( \frac{x^3}{x} = x^2 \)
• \( \frac{x^2}{x} = x \)

These steps simplify the function considerably, paving the way for easier differentiation. Always aim to simplify functions wherever possible, as this often reduces errors in calculation during subsequent steps.

Differentiation

Differentiation is one of the cornerstones of calculus. It involves finding the derivative, which represents the rate of change of a function. Simply put, the derivative of a function \( f(x) \) is noted as \( f'(x) \).
Once you have a simplified function, differentiation becomes a straightforward process. With our simplified function \( f(x) = x^2 + x \), the aim is to determine how this function changes as \( x \) changes. This step is directly linked to understanding the slope of the curve represented by the function.
It's crucial to apply the right differentiation rules. Whether dealing with polynomial expressions, trigonometric functions, or others, using the appropriate rule ensures you're calculating accurately.

Power Rule

The power rule is a fundamental technique in calculus for differentiating functions of the form \( x^n \), where \( n \) is any real number. This rule is particularly useful for polynomial functions.
The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \). Applying this to our function \( f(x) = x^2 + x \):

• For \( x^2 \), the power is 2, so the derivative is \( 2x^{2-1} = 2x \).
• For \( x \), which is \( x^1 \), the derivative is simply 1 because \( 1x^{1-1} = 1 \).

Thus, the derivative for the entire function \( f(x) = x^2 + x \) becomes \( f'(x) = 2x + 1 \).
Understanding and applying the power rule correctly is essential for anyone working with calculus, as it simplifies differentiation dramatically and is frequently used in various mathematical analyses.