Derivative Of 1 3x 3

Understanding the Derivative of 1/(3x + 3)

This article delves into the intricacies of finding the derivative of the function f(x) = 1/(3x + 3), a crucial concept in calculus. We will explore various methods, provide a step-by-step solution, and explain the underlying mathematical principles. Understanding this seemingly simple derivative unlocks a deeper understanding of differentiation and its applications in various fields. This comprehensive guide is designed for students learning calculus, as well as anyone seeking a refresher on differentiation techniques.

Introduction: Derivatives and their Significance

The derivative of a function measures its instantaneous rate of change. In simpler terms, it tells us how much the function's output changes in response to a tiny change in its input. This concept is fundamental in understanding slopes of curves, optimization problems, and many other applications across science and engineering. For the function f(x) = 1/(3x + 3), finding the derivative will allow us to analyze its slope at any given point on its graph.

Method 1: The Power Rule and Chain Rule

This method uses two fundamental rules of differentiation: the power rule and the chain rule.

• Power Rule: The derivative of xⁿ is nx^n-1.
• Chain Rule: The derivative of f(g(x)) is f'(g(x)) * g'(x).

First, we rewrite the function to make it easier to apply these rules:

f(x) = (3x + 3)^-1

Now we apply the chain rule, where f(u) = u^-1 and u = 3x + 3.

1. Find the derivative of the outer function: The derivative of u^-1 with respect to u is -u^-2.

2. Find the derivative of the inner function: The derivative of 3x + 3 with respect to x is 3.

3. Apply the chain rule: Multiply the derivatives of the outer and inner functions:

   f'(x) = - (3x + 3)^-2 * 3

4. Simplify:

   f'(x) = -3 / (3x + 3)²

   We can further simplify this by factoring out a 3 from the denominator:

   f'(x) = -1 / (x + 1)²

Method 2: The Quotient Rule

The quotient rule provides another approach for finding the derivative of a function in the form of a fraction. The quotient rule states:

If f(x) = g(x) / h(x), then f'(x) = [h(x)g'(x) - g(x)h'(x)] / [h(x)]²

In our case, g(x) = 1 and h(x) = 3x + 3. Therefore:

1. Find the derivative of g(x): g'(x) = 0 (the derivative of a constant is zero).

2. Find the derivative of h(x): h'(x) = 3 (the derivative of 3x + 3 with respect to x is 3).

3. Apply the quotient rule:

   f'(x) = [(3x + 3)(0) - (1)(3)] / (3x + 3)²

4. Simplify:

   f'(x) = -3 / (3x + 3)²

   As before, simplifying further yields:

   f'(x) = -1 / (x + 1)²

Detailed Explanation of the Steps

Let's break down the simplification process in more detail:

The expression -3 / (3x + 3)² can be simplified by factoring out a 3 from the denominator:

(3x + 3)² = [3(x + 1)]² = 9(x + 1)²

Substituting this back into the derivative expression, we get:

-3 / [9(x + 1)²] = -1 / [3(x + 1)²]

This simplification might seem minor, but it's crucial for expressing the derivative in its most concise and easily interpretable form.

The Significance of the Negative Sign

The negative sign in the final derivative, f'(x) = -1 / (x + 1)², is significant. It indicates that the function f(x) = 1/(3x + 3) is always decreasing for values of x where it's defined (x ≠ -1). The slope of the tangent line to the curve at any point is negative.

Analyzing the Derivative

The derivative f'(x) = -1 / (x + 1)² tells us a great deal about the behavior of the original function:

• Undefined at x = -1: The derivative is undefined at x = -1 because the denominator becomes zero. This corresponds to a vertical asymptote in the original function's graph.

• Always Negative: As mentioned earlier, the negative sign means the function is always decreasing where it's defined.

• Approaches Zero as x Approaches Infinity: As x gets very large (either positive or negative), the derivative approaches zero. This implies that the function's rate of change becomes smaller and smaller as we move further away from the vertical asymptote.

Applications of the Derivative

The derivative of 1/(3x + 3) finds applications in various fields:

• Physics: Analyzing rates of change in physical systems, like the decay of a radioactive substance or the cooling of an object.

• Economics: Modeling changes in economic quantities such as supply, demand, or cost functions.

• Engineering: Designing optimal systems, analyzing the performance of machines, and predicting system behavior.

Frequently Asked Questions (FAQ)

Q: Why is the chain rule necessary in this problem?

A: The chain rule is necessary because we are differentiating a composite function. The function 1/(3x + 3) can be seen as a function of another function: The outer function is 1/u and the inner function is u = 3x + 3. The chain rule allows us to differentiate composite functions by differentiating the outer and inner functions separately and multiplying the results.

Q: Can we use other differentiation techniques?

A: Yes, while the power rule and chain rule, and the quotient rule provide straightforward approaches, other methods exist. However, these methods would likely involve more complex algebraic manipulations without providing significant advantages.

Q: What is the significance of the vertical asymptote at x = -1?

A: The vertical asymptote at x = -1 indicates that the function approaches infinity as x approaches -1 from either side. This signifies a discontinuity in the function, where the function is undefined at that point.

Q: How can I check my answer?

A: You can use a graphing calculator or online calculus tools to plot the original function and its derivative. You can visually confirm that the derivative's sign matches the function's behavior (increasing or decreasing) and that the derivative is undefined at the vertical asymptote. You can also use numerical methods to approximate the derivative at different points and compare them to the analytical derivative we calculated.

Conclusion: A Deeper Understanding of Differentiation

This comprehensive exploration of the derivative of 1/(3x + 3) highlights the power and versatility of calculus. By applying fundamental rules like the power rule, chain rule, and quotient rule, we have successfully derived the expression for the instantaneous rate of change of this function. Understanding this process not only helps us solve specific problems but also deepens our understanding of how functions behave and how their derivatives reflect this behavior. Remember that the key to mastering calculus lies in understanding the underlying principles and applying them methodically. Through practice and careful analysis, you will build a solid foundation in differentiation and be well-equipped to tackle more complex problems.