Problem 5 - The Calculus with Analytic Geometry

Question

Find the derivative of the given function. $$ f(x)=4 x^{1 / 2}+5 x^{-1 / 2} $$

Step-by-Step Solution

Answer

\( f'(x) = 2 x^{-1/2} - \frac{5}{2} x^{-3/2} \).

Step 1: Identify the function parts

The given function is composed of two terms: \( f(x) = 4 x^{1/2} + 5 x^{-1/2} \). Identify each term individually for differentiation.

Step 2: Differentiate the first term

For the first term, \( 4 x^{1/2} \), use the power rule of differentiation which states \( \frac{d}{dx}[x^n] = nx^{n-1} \). Here, \( n = 1/2 \): \( \frac{d}{dx}[4 x^{1/2}] = 4 \cdot \frac{1}{2} x^{1/2-1} = 2 x^{-1/2} \).

Step 3: Differentiate the second term

For the second term, \( 5 x^{-1/2} \), use the power rule of differentiation with \( n = -1/2 \): \( \frac{d}{dx}[5 x^{-1/2}] = 5 \cdot (-1/2) x^{-1/2-1} = -\frac{5}{2} x^{-3/2} \).

Step 4: Sum the derivatives

Combine the individual derivatives to find the overall derivative of the function: \( f'(x) = 2 x^{-1/2} - \frac{5}{2} x^{-3/2} \).

Key Concepts

The Power Rule

The power rule is a fundamental tool in calculus for finding the derivative of functions. Its formula is simple yet powerful: if you have a term in the form of \(x^n\), the derivative is found by bringing the exponent down as a coefficient and then subtracting one from the exponent. Mathematically, this is expressed as: \[ \frac{d}{dx}[x^n] = nx^{n-1} \] Let's break it down:

• First, the exponent \(n\) becomes a coefficient placed in front of the term.
• Next, subtract 1 from the original exponent to get your new exponent.

This rule applies to any real number exponent. In our example function, \(f(x) = 4 x^{1/2} + 5 x^{-1/2}\), we applied the power rule with different exponents (\(1/2\) and \(-1/2\)), transforming each term independently.

Step-by-step Differentiation

Differentiation is the process of finding the derivative of a function. To differentiate a function step-by-step, especially when dealing with multiple terms, break down the function into simpler parts. Here are the steps we followed for the function \(f(x) = 4 x^{1/2} + 5 x^{-1/2}\):

• **Step 1: Identify each term individually.**
  The function has two parts: \(4 x^{1/2}\) and \(5 x^{-1/2}\). Focus on one term at a time.
• **Step 2: Apply the power rule to each term.**
  For the first term \(4 x^{1/2}\), the power rule gives us \(2 x^{-1/2}\).
  For the second term \(5 x^{-1/2}\), the power rule gives us \(-\frac{5}{2} x^{-3/2}\).
• **Step 3: Sum the derivatives.**
  Combine the individual derivatives to form: \(f'(x) = 2 x^{-1/2} - \frac{5}{2} x^{-3/2}\).

By tackling each term one by one and applying the power rule, differentiation becomes a straightforward process.

Combining Derivatives

Once you have found the derivatives of individual terms in a function, the next step is to combine them. This is a crucial part of differentiation because it gives us the overall derivative of the original function.

Consider our function: \(f(x) = 4 x^{1/2} + 5 x^{-1/2}\). After applying the power rule, we obtained:

• Derivative of \(4 x^{1/2}\) is \(2 x^{-1/2}\)
• Derivative of \(5 x^{-1/2}\) is \(-\frac{5}{2} x^{-3/2}\)

The final step is to combine these parts. Therefore:
\[ f'(x) = 2 x^{-1/2} - \frac{5}{2} x^{-3/2} \]
Combining derivatives is simple arithmetic. You just place each differential term you calculated into a single expression. This method ensures your final derivative represents the rate of change for the entire function.

To recap: Combine the derivatives of parts geometrically by adding or subtracting them as needed. This final expression provides the complete solution to your differentiation exercise.