Problem 34 - Single Variable Calculus

Question

Let \(f(x)=4 x^{5 / 4}+2 x^{3 / 2}+x\). Find a. \(f^{\prime}(0)\) b. \(f^{\prime}(16)\)

Step-by-Step Solution

Answer

a. \(f'(0) = 1\) b. \(f'(16) = 23\)

Step 1: Find the first derivative of the function

To find the derivative of the given function, we will apply the power rule of differentiation, which states that if \(f(x) = x^n\), then \(f'(x) = n \cdot x^{n-1}\). Let's apply the rule to each term of the function. For the term \(4x^{5/4}\), the power rule gives: \(4 \cdot \frac{5}{4}x^{\frac{5}{4} - 1} = 5x^{1/4}\). For the term \(2x^{3/2}\), the power rule gives: \(2 \cdot \frac{3}{2}x^{\frac{3}{2} - 1} = 3x^{1/2}\). For the term \(x^1\), the power rule gives: \(1 \cdot x^{1 - 1} = 1\). Putting it all together, we get the first derivative: \(f'(x) = 5x^{1/4} + 3x^{1/2} + 1\).

Step 2: Compute \(f'(0)\)

Let's compute the value of the first derivative at \(x = 0\). \(f'(0) = 5(0)^{1/4} + 3(0)^{1/2} + 1 = 5(0) + 3(0) + 1 = 0 + 0 + 1 = 1\). So, \(f'(0)= 1\).

Step 3: Compute \(f'(16)\)

Now, let's compute the value of the first derivative at \(x = 16\). \(f'(16) = 5(16)^{1/4} + 3(16)^{1/2} + 1 = 5(2) + 3(4) + 1 = 10 + 12 + 1 = 23\). So, \(f'(16)= 23\). To summarize the results: a. \(f'(0) = 1\) b. \(f'(16) = 23\)

Key Concepts

Derivative of a Function

Understanding the derivative of a function is crucial in calculus. It gives us the rate at which a function is changing at any given point. Think of it like the speedometer of a car—it tells you how fast you're going at a specific instant.

In formal terms, the derivative of a function at a point is the limit of the average rate of change as the interval over which we calculate the rate of change approaches zero. In simpler words, if you have a function describing the position of a car over time, the derivative tells you the car's instantaneous speed at any moment.

Mathematically, if we have a function, say, \( f(x) \), its derivative is notated as \( f'(x) \) or \( \frac{df}{dx} \), and is found by applying derivative rules such as the power rule, product rule, quotient rule, or chain rule, depending on the form of the function.

Applying the Power Rule

The power rule is one of the basic and most frequently used rules for finding derivatives. It applies to functions where the variable, typically \( x \), is raised to a power, such as \( x^n \).

The power rule simply states that to find the derivative of \( x^n \), you multiply the power by the coefficient of \( x \), and then decrease the power by 1. The rule is elegantly simple: \( \frac{d}{dx}[x^n] = nx^{n-1} \).

For example, using the power rule, the derivative of \( 2x^3 \) would be \( 3 \times 2x^{2} = 6x^2 \). This rule streamlines the process of differentiation by providing an almost immediate result for any power of \( x \).

Computing Derivatives at Specific Points

In many practical situations, you're not just interested in the derivative itself, but its value at a particular point. For instance, you might want to know the slope of a curve at \( x=5 \) or the rate of change of a quantity when \( t=10 \).

To compute a derivative at a specific point:\( x=a \), find the general form of the derivative, and then replace \( x \) with \( a \). If the derivative is \( f'(x) \), then \( f'(a) \) gives the slope or rate of change at that particular value of \( a \).

Example:

If \( f(x) = x^2 \), the general derivative is \( f'(x) = 2x \). To find the slope at \( x=3 \), simply compute \( f'(3) = 2 \times 3 = 6 \). This number represents the steepness of the curve at that point.

First Derivative Computation

The first step in many calculus problems is computing the first derivative. The first derivative reflects the function's rate of change and provides insights into the function's behavior—whether it's increasing or decreasing and where it might have peaks or troughs.

To compute the first derivative, apply the appropriate differentiation rules to each term of the function separately and then combine them. Remember that constants become zero when differentiated, and the derivative of the variable \( x \) with respect to itself is 1.

After finding the first derivative, \( f'(x) \), you can use it to compute slopes at specific points or to find critical values where the slope is zero (potentially indicating a maximum or minimum point on the graph). Analysis of the first derivative is foundational for studying the motion of objects, optimizing functions, and understanding rates of change in the physical sciences, economics, and beyond.