How to Solve for the Slope of the Curve y2 3x at the Point (1/30, -1/30) Using the Formal Definition of Derivative

Calculus is a powerful tool for understanding the behavior of functions and their rates of change. One of the fundamental concepts in calculus is the derivative, which allows us to find the slope of a curve at any given point. In this article, we will explore how to determine the slope of the curve y2 3x at the specific point (1/30, -1/30) using the formal definition of a derivative. Understanding this process is crucial for students and professionals in fields such as mathematics, physics, and engineering.

1. Rewriting the Equation in Terms of y

First, let's start by rewriting the given equation y2 3x in a form that makes it easier to apply the derivative concept.

From the given equation:

y2 3x

Isolate x to express it in terms of y:

x 1/3 * y2 - 1/3

2. Differentiating with Respect to y

Next, we'll differentiate the equation with respect to y to find the derivative. The derivative of the function gives us the slope of the tangent line at any point on the curve.

Take the derivative of x 1/3 * y2 - 1/3 with respect to y:

dy/dx (2/3) * y

It's important to note that this step is slightly different from the initial approach mentioned in the original text. Here, we're finding the derivative with respect to y, and the relationship is expressed as dy/dx (the rate of change of x with respect to y).

3. Evaluating the Slope at the Point (1/30, -1/30)

Now that we have the derivative, let's evaluate it at the specific point (1/30, -1/30).

Plug in the value y -1/30 into the derivative equation:

dx/dy (2/3) * (-1/30)

Perform the calculation:

dx/dy -2/90 -1/45

This value, -1/45, represents the slope of the curve y2 3x at the point (1/30, -1/30).

4. Understanding the Importance of the Derivative

Let's delve deeper into the significance of the derivative in this context:

Physical Interpretation: The derivative provides the rate of change of one variable with respect to another. Here, it tells us how x changes with respect to y at the specific point.

Geometric Interpretation: The value of the derivative at a given point on a curve gives the slope of the tangent line to the curve at that point. In this case, the curve is y2 3x, and the slope is -1/45 at the point (1/30, -1/30).

5. Summary and Conclusion

In conclusion, we have successfully used the formal definition of the derivative to find the slope of the curve y2 3x at the point (1/30, -1/30). The step-by-step process involves:

Rewriting the equation in terms of y Computing the derivative with respect to y Evaluating the derivative at the specific point

This method not only helps in understanding the curve's behavior but also reinforces the utility of derivatives in analyzing complex functions. For students and professionals alike, mastering the concept of derivatives is essential for tackling more advanced problems in calculus and related fields.

Always keep learning, Joe!