Find The Tangent Line Slope To -3x² + 4xy + Y³ = -8 At (4, 2)

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Introduction

In the realm of calculus, one of the fundamental concepts is the tangent line to a curve at a given point. The slope of this tangent line provides valuable information about the curve's behavior at that specific location, such as its rate of change. In this comprehensive guide, we will delve into the process of finding the slope of the tangent line to the curve defined by the equation $-3x^2 + 4xy + y^3 = -8$ at the point $(4, 2)$. This problem requires us to use the technique of implicit differentiation, a powerful tool for finding derivatives of functions that are not explicitly defined in terms of one variable. Understanding implicit differentiation and its application in finding tangent lines is crucial for various applications in mathematics, physics, and engineering. We will break down the steps involved, providing clear explanations and examples to ensure a thorough understanding of the method.

Understanding Implicit Differentiation

Before we dive into the specifics of our problem, it's essential to grasp the concept of implicit differentiation. Unlike explicit functions, where one variable is directly expressed in terms of another (e.g., $y = f(x)$), implicit functions define a relationship between variables without explicitly isolating one. For instance, our equation $-3x^2 + 4xy + y^3 = -8$ is an implicit function because neither $x$ nor $y$ is isolated. To find the derivative $\frac{dy}{dx}$ in such cases, we use implicit differentiation. This technique involves differentiating both sides of the equation with respect to $x$, treating $y$ as a function of $x$ and applying the chain rule where necessary. The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In the context of implicit differentiation, this means that whenever we differentiate a term involving $y$, we must multiply by $\frac{dy}{dx}$ to account for the fact that $y$ is a function of $x$. The process might seem complex at first, but with practice, it becomes a straightforward and indispensable tool in calculus.

Step-by-Step Solution

Now, let's apply the principles of implicit differentiation to find the slope of the tangent line to the curve $-3x^2 + 4xy + y^3 = -8$ at the point $(4, 2)$.

Step 1: Differentiate Both Sides

The first step is to differentiate both sides of the equation with respect to $x$. This requires careful application of the power rule, product rule, and chain rule. Remember that when differentiating terms involving $y$, we must multiply by $\frac{dy}{dx}$. Let's break down the differentiation process:

  • Differentiating $-3x^2$: Applying the power rule, we get $-6x$.
  • Differentiating $4xy$: This term requires the product rule, which states that the derivative of $uv$ is $u'v + uv'$. Here, $u = 4x$ and $v = y$. So, the derivative is $4y + 4x\frac{dy}{dx}$.
  • Differentiating $y^3$: Using the power rule and chain rule, we get $3y^2\frac{dy}{dx}$.
  • Differentiating $-8$: The derivative of a constant is always $0$.

Putting it all together, the differentiated equation looks like this:

−6x+4y+4xdydx+3y2dydx=0-6x + 4y + 4x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0

Step 2: Isolate $\frac{dy}{dx}$

Our next goal is to isolate $\frac{dy}{dx}$ on one side of the equation. This involves rearranging the terms so that all terms containing $\frac{dy}{dx}$ are on one side, and all other terms are on the other side. From the differentiated equation, we have:

−6x+4y+4xdydx+3y2dydx=0-6x + 4y + 4x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0

Move the terms without $\frac{dy}{dx}$ to the right side:

4xdydx+3y2dydx=6x−4y4x\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 6x - 4y

Now, factor out $\frac{dy}{dx}$ from the left side:

dydx(4x+3y2)=6x−4y\frac{dy}{dx}(4x + 3y^2) = 6x - 4y

Finally, divide both sides by $(4x + 3y^2)$ to isolate $\frac{dy}{dx}$:

dydx=6x−4y4x+3y2\frac{dy}{dx} = \frac{6x - 4y}{4x + 3y^2}

This expression gives us the slope of the tangent line at any point $(x, y)$ on the curve.

Step 3: Substitute the Point $(4, 2)$

Now that we have an expression for $\frac{dy}{dx}$, we can find the slope of the tangent line at the specific point $(4, 2)$. Substitute $x = 4$ and $y = 2$ into the equation:

dydx=6(4)−4(2)4(4)+3(2)2\frac{dy}{dx} = \frac{6(4) - 4(2)}{4(4) + 3(2)^2}

Simplify the expression:

dydx=24−816+12\frac{dy}{dx} = \frac{24 - 8}{16 + 12}

dydx=1628\frac{dy}{dx} = \frac{16}{28}

dydx=47\frac{dy}{dx} = \frac{4}{7}

Therefore, the slope of the tangent line to the curve at the point $(4, 2)$ is $\frac{4}{7}$. This means that at the point $(4, 2)$, the curve is increasing with a slope of $\frac{4}{7}$. The tangent line at this point represents the instantaneous rate of change of the curve, and its slope provides valuable information about the curve's behavior in the vicinity of the point.

Visualizing the Tangent Line

To further solidify our understanding, it's helpful to visualize the tangent line. Imagine the curve $-3x^2 + 4xy + y^3 = -8$ plotted on a graph. At the point $(4, 2)$, draw a line that touches the curve at that point and has a slope of $\frac{4}{7}$. This line is the tangent line we've just calculated. It represents the best linear approximation of the curve at that point. Visualizing the tangent line helps to connect the abstract concept of a derivative to the geometric interpretation of a slope.

Applications of Tangent Lines

Understanding tangent lines and their slopes is not just a theoretical exercise; it has numerous practical applications in various fields. In physics, for example, the tangent line can represent the instantaneous velocity of an object moving along a curved path. The slope of the tangent line at a given time gives the object's speed and direction at that moment. In engineering, tangent lines are used in optimization problems, such as finding the maximum or minimum value of a function. The points where the tangent line is horizontal (i.e., has a slope of zero) often correspond to local maxima or minima of the function. In economics, tangent lines can be used to model marginal cost or marginal revenue, which are important concepts in understanding the profitability of a business. The slope of the tangent line in these contexts represents the additional cost or revenue generated by producing one more unit of a product or service. These are just a few examples of how the concept of tangent lines and their slopes plays a crucial role in understanding and solving real-world problems.

Common Mistakes to Avoid

When working with implicit differentiation, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and ensure accurate results.

  • Forgetting the Chain Rule: One of the most frequent mistakes is forgetting to apply the chain rule when differentiating terms involving $y$. Remember that $y$ is a function of $x$, so the derivative of $y^n$ with respect to $x$ is $ny^{n-1}\frac{dy}{dx}$, not just $ny^{n-1}$. This is a critical step in implicit differentiation, and overlooking it will lead to incorrect results.
  • Incorrectly Applying the Product Rule: When differentiating terms like $4xy$, it's essential to apply the product rule correctly. The derivative of $uv$ is $u'v + uv'$, so the derivative of $4xy$ is $4y + 4x\frac{dy}{dx}$, not just $4x\frac{dy}{dx}$ or $4y$. Make sure to include both terms in the product rule.
  • Algebraic Errors: After differentiating, isolating $\frac{dy}{dx}$ involves algebraic manipulation. Mistakes in this step, such as incorrectly factoring or dividing, can lead to a wrong expression for $\frac{dy}{dx}$. Double-check your algebraic steps to ensure accuracy.
  • Substituting Before Differentiating: It's crucial to differentiate the equation before substituting the point $(x, y)$. Substituting too early will eliminate $\frac{dy}{dx}$ from the equation, making it impossible to solve for the slope of the tangent line. Differentiate first, then substitute.
  • Not Simplifying the Final Answer: While it's technically correct to leave the answer as an unsimplified fraction, it's good practice to simplify the result as much as possible. This not only makes the answer cleaner but also reduces the chance of making errors in subsequent calculations. Simplify fractions and combine like terms whenever possible.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving implicit differentiation problems.

Conclusion

In this comprehensive guide, we've walked through the process of finding the slope of the tangent line to the curve $-3x^2 + 4xy + y^3 = -8$ at the point $(4, 2)$. We began by understanding the concept of implicit differentiation, a powerful technique for finding derivatives of implicitly defined functions. We then applied this technique step-by-step, differentiating both sides of the equation, isolating $\frac{dy}{dx}$, and substituting the given point to find the slope. The result, $\frac{4}{7}$, represents the slope of the tangent line at the point $(4, 2)$. We also discussed the importance of visualizing the tangent line and its applications in various fields, as well as common mistakes to avoid when using implicit differentiation. By mastering this technique, you'll be well-equipped to tackle a wide range of calculus problems involving implicit functions and tangent lines. The ability to find the slope of a tangent line is a fundamental skill in calculus with wide-ranging applications in mathematics, physics, engineering, and economics. This guide has provided a detailed explanation of the process, equipping you with the knowledge and tools to confidently solve similar problems.