Finding The Derivative Of F(x) = (2x² + 3x - 4) / (13x + 6) A Step-by-Step Guide

Introduction: Delving into the Realm of Derivatives

In the fascinating world of calculus, derivatives stand as fundamental tools for understanding the rate at which functions change. They provide insights into the slope of a curve at any given point, the velocity of a moving object, and countless other dynamic processes. When faced with a function like f(x) = (2x² + 3x - 4) / (13x + 6), finding its derivative, denoted as f'(x), becomes a crucial step in unraveling its behavior. In this comprehensive exploration, we will embark on a step-by-step journey to determine the derivative of this rational function, employing the powerful quotient rule and showcasing the intricacies of calculus.

Understanding the Significance of Derivatives

Before we plunge into the mechanics of finding the derivative, it's essential to grasp the significance of this mathematical operation. The derivative, at its core, represents the instantaneous rate of change of a function. Imagine a graph of f(x); the derivative at a particular point gives us the slope of the line tangent to the curve at that exact location. This concept has far-reaching applications across various fields, including physics, engineering, economics, and computer science. For instance, in physics, the derivative of a position function with respect to time yields the velocity, while the derivative of velocity gives acceleration. In economics, derivatives are used to optimize production costs and analyze market trends. The versatility of derivatives makes them indispensable tools in mathematical analysis and problem-solving.

The Quotient Rule: A Gateway to Differentiating Rational Functions

Our function, f(x) = (2x² + 3x - 4) / (13x + 6), is a rational function, a ratio of two polynomials. To tackle the differentiation of such functions, we turn to the quotient rule. This rule provides a systematic way to find the derivative of a function that is expressed as the quotient of two other functions. The quotient rule states that if we have a function h(x) = u(x) / v(x), where u(x) and v(x) are differentiable functions, then the derivative h'(x) is given by:

h'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]²

In simpler terms, the derivative of a quotient is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. This formula might seem a bit daunting at first, but with practice, it becomes a straightforward tool for differentiating rational functions.

Applying the Quotient Rule to f(x) = (2x² + 3x - 4) / (13x + 6)

Now, let's apply the quotient rule to our function f(x) = (2x² + 3x - 4) / (13x + 6). We can identify the numerator as u(x) = 2x² + 3x - 4 and the denominator as v(x) = 13x + 6. The first step is to find the derivatives of both u(x) and v(x).

Finding the Derivatives of u(x) and v(x)

To find u'(x), the derivative of u(x) = 2x² + 3x - 4, we use the power rule, which states that the derivative of x^n is nx^(n-1). Applying this rule, we get:

u'(x) = 4x + 3

Similarly, to find v'(x), the derivative of v(x) = 13x + 6, we again use the power rule, along with the constant rule (the derivative of a constant is zero):

v'(x) = 13

With the derivatives of u(x) and v(x) in hand, we are ready to plug these into the quotient rule formula.

Implementing the Quotient Rule Formula

Recall the quotient rule formula:

f'(x) = [v(x) * u'(x) - u(x) * v'(x)] / [v(x)]²

Substituting u(x), v(x), u'(x), and v'(x) into the formula, we get:

f'(x) = [(13x + 6) * (4x + 3) - (2x² + 3x - 4) * 13] / (13x + 6)²

This expression represents the derivative of f(x). However, to obtain a more simplified and manageable form, we need to perform some algebraic manipulations.

Simplifying the Derivative: A Journey into Algebraic Manipulation

The expression we obtained for f'(x) is correct, but it's not in its simplest form. To make it more readable and easier to work with, we need to expand the numerator and simplify the resulting expression. This involves multiplying out the terms and combining like terms.

Expanding the Numerator: Unveiling the Terms

The numerator of our derivative is [(13x + 6) * (4x + 3) - (2x² + 3x - 4) * 13]. Let's expand the first part, (13x + 6) * (4x + 3), using the distributive property:

(13x + 6) * (4x + 3) = 13x * 4x + 13x * 3 + 6 * 4x + 6 * 3 = 52x² + 39x + 24x + 18 = 52x² + 63x + 18

Next, let's expand the second part, (2x² + 3x - 4) * 13:

(2x² + 3x - 4) * 13 = 26x² + 39x - 52

Now, we can substitute these expanded expressions back into the numerator:

f'(x) = [52x² + 63x + 18 - (26x² + 39x - 52)] / (13x + 6)²

Combining Like Terms: The Art of Simplification

The next step is to combine like terms in the numerator. This involves grouping together terms with the same power of x:

f'(x) = [52x² + 63x + 18 - 26x² - 39x + 52] / (13x + 6)²

f'(x) = [(52x² - 26x²) + (63x - 39x) + (18 + 52)] / (13x + 6)²

f'(x) = [26x² + 24x + 70] / (13x + 6)²

We have now successfully simplified the numerator. The derivative, f'(x), is expressed as a rational function with a quadratic polynomial in the numerator and the square of a linear polynomial in the denominator.

The Final Derivative: f'(x) = (26x² + 24x + 70) / (13x + 6)²

After applying the quotient rule, expanding the terms, and simplifying the expression, we arrive at the final derivative of f(x) = (2x² + 3x - 4) / (13x + 6):

f'(x) = (26x² + 24x + 70) / (13x + 6)²

This expression represents the instantaneous rate of change of the function f(x) at any point x. It provides valuable insights into the behavior of the function, such as its increasing and decreasing intervals, critical points, and concavity. The derivative is a powerful tool for analyzing and understanding the function's characteristics.

Applications of the Derivative: Beyond the Formula

The derivative we have found is not just an abstract mathematical expression; it has numerous practical applications. Let's explore some of the ways in which this derivative can be used:

Finding Critical Points: Identifying Maxima and Minima

Critical points are points where the derivative of a function is either zero or undefined. These points are crucial in identifying the local maxima and minima of the function, which represent the highest and lowest points in a given interval. To find the critical points of f(x), we need to solve the equation f'(x) = 0:

(26x² + 24x + 70) / (13x + 6)² = 0

A rational function is zero when its numerator is zero. Therefore, we need to solve the quadratic equation:

26x² + 24x + 70 = 0

This quadratic equation can be solved using the quadratic formula or by factoring. The solutions will give us the x-coordinates of the critical points.

Determining Intervals of Increase and Decrease: Understanding Function Behavior

The sign of the derivative tells us whether the function is increasing or decreasing. If f'(x) > 0 in an interval, then f(x) is increasing in that interval. Conversely, if f'(x) < 0, then f(x) is decreasing. To determine the intervals of increase and decrease, we can analyze the sign of f'(x) in different intervals defined by the critical points and any points where the denominator of f'(x) is zero.

Analyzing Concavity: Unveiling the Shape of the Curve

The second derivative, f''(x), provides information about the concavity of the function. If f''(x) > 0 in an interval, then f(x) is concave up, meaning it curves upwards. If f''(x) < 0, then f(x) is concave down, meaning it curves downwards. To analyze concavity, we would need to find the second derivative of f(x), which involves differentiating f'(x).

Conclusion: The Power of Derivatives in Mathematical Analysis

In this exploration, we have successfully navigated the process of finding the derivative of the rational function f(x) = (2x² + 3x - 4) / (13x + 6). We employed the quotient rule, a fundamental tool in calculus, and demonstrated the importance of algebraic manipulation in simplifying the resulting expression. The final derivative, f'(x) = (26x² + 24x + 70) / (13x + 6)², provides valuable insights into the function's behavior, including its critical points, intervals of increase and decrease, and concavity. Derivatives are indispensable tools in mathematical analysis, enabling us to understand and model dynamic processes in various fields. Mastering the techniques of differentiation opens doors to a deeper understanding of the mathematical world and its applications.

Keywords: derivatives, quotient rule, differentiation, rational functions, rate of change, critical points, intervals of increase and decrease, concavity, mathematical analysis