Problem 15 - Brief Applied Calculus

Question

Find the derivative of each function. $$ f(x)=\frac{1}{x^{1 / 2}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -\frac{1}{2x^{3/2}} \).

Step 1: Rewrite the function

First, express the function in a form that makes it easier to differentiate. The function given is \( f(x) = \frac{1}{x^{1/2}} \). This can be rewritten using negative exponents as \( f(x) = x^{-1/2} \).

Step 2: Apply the power rule for differentiation

To find the derivative of \( f(x) = x^{-1/2} \), use the power rule which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Here, the power \( n = -1/2 \). Apply the rule: \[ f'(x) = -\frac{1}{2}x^{-1/2 - 1} = -\frac{1}{2}x^{-3/2} \].

Step 3: Simplify the derivative

Finally, express the derivative in a more conventional form if needed. Since \( x^{-3/2} \) can be rewritten using radicals, the simplified form of the derivative is \[ f'(x) = -\frac{1}{2} \cdot \frac{1}{x^{3/2}} = -\frac{1}{2x^{3/2}} \].

Key Concepts

Derivatives

In calculus, derivatives represent the rate at which a function is changing at any given point. It's like understanding the slope of the tangent line at a particular point on a curve. Derivatives are foundational for many advanced topics in calculus, as they model dynamic changes. When you see a function, such as \( f(x) = x^{-1/2} \), taking its derivative will tell you how this function behaves as \( x \) changes.
To find a derivative, one can apply various rules and techniques based on the form of the function. In simpler terms, derivatives help us understand things like velocity if we think of the function as representing position over time.
Derivatives can be represented by \( f'(x) \), or \( \frac{df}{dx} \), indicating the differentiation with respect to x.

Power Rule

The power rule is a straightforward method in calculus for finding the derivative of functions in the form of \( x^n \). This rule is efficient and significantly cuts down the complexity involved in differentiation.
Here's the power rule: If you have \( f(x) = x^n \), then the derivative \( f'(x) \) is given by \( nx^{n-1} \). This involves multiplying by the power and decreasing the power by one. For example, for \( f(x) = x^{-1/2} \), the derivative is found by multiplying by \(-1/2 \) and then reducing the power to \(-1/2 - 1\).
**Key Points of Power Rule**:

• Easy to use for monomial terms.
• Only works for powers of \( x \).
• Makes differentiation of simple polynomials quick and efficient.

Using the power rule makes it much simpler to calculate derivatives, especially for polynomial functions or functions that can be rewritten in a polynomial-like form.

Negative Exponents

Exponents are powerful mathematical tools, and understanding negative exponents is crucial for working with certain calculus problems. A negative exponent indicates that the number is a reciprocal. For instance, \( x^{-n} = \frac{1}{x^n} \). This transformation allows for easier manipulation and differentiation of functions.
In the context of calculus, negative exponents allow us to rewrite fractions into forms where differentiation rules, like the power rule, can be applied more easily. For example, converting \( \frac{1}{x^{1/2}} \) to \( x^{-1/2} \) simplifies the process of finding derivatives.
**Advantages of Using Negative Exponents**:

• Simplifies expressions by turning fractions into workable power terms.
• Enables the application of differentiation techniques like the power rule.
• Makes derivatives clearer and often easier to simplify further.

By mastering negative exponents, students can enhance their problem-solving skills in calculus, especially when dealing with more complex fractional expressions.