What is the Slope of a Curve at a Given Point?

Implicit differentiation is a powerful technique in calculus used to find the slope of a curve defined by a non-linear equation. In this article, we will explore the process of finding the derivative, and hence the slope, of a specific curve at a given point using implicit differentiation. Let's consider the curve defined by the equation:

The Curve Equation and Implicit Differentiation

The given equation of the curve is:

$$y^2x^2y^4-2x$$

To find the derivative of this curve with respect to x, we'll use implicit differentiation.

Step-by-Step Example: Finding the Slope at Point (3, 4)

To find the slope of the curve at the point (3, 4), we follow these steps:

Step 1: Differentiate with Respect to x

First, we differentiate both sides of the equation with respect to x:

$$frac{d}{dx}(y^2) frac{d}{dx}(x^2) frac{d}{dx}(y^4) - frac{d}{dx}(2x)$$

Using the chain rule and the product rule, we get:

$$2y frac{dy}{dx} 2x 4y^3 frac{dy}{dx} - 2$$

Step 2: Rearrange the Equation

Next, we rearrange the equation to isolate dy/dx:

$$2y frac{dy}{dx} - 4y^3 frac{dy}{dx} -2 - 2x$$

Factoring out dy/dx:

$$(2y - 4y^3) frac{dy}{dx} -2 - 2x$$

Step 3: Solve for dy/dx

Solving for dy/dx:

$$frac{dy}{dx} frac{-2 - 2x}{2y - 4y^3}$$

Step 4: Substitute the Point (3, 4)

Now, substitute the point (3, 4) into the equation:

$$frac{dy}{dx} frac{-2 - 2(3)}{2(4) - 4(4^3)}$$

Calculate the value:

$$$-2 - 6 -8$$ $$$2(4) - 4(64) 8 - 256 -248$$ $$frac{dy}{dx} frac{-8}{-248} frac{8}{248} frac{1}{31}$$

Thus, the slope of the curve at the point (3, 4) is 1/31.

Understanding the Process and Common Pitfalls

It's important to note that not all points on the curve satisfy the equation. In the example where we try to find the slope at the point (3, 4) using calculus techniques, we must first verify that the point lies on the curve. If the point does not lie on the curve, as in the example below, the slope calculation does not make sense.

Checking if a Point Lies on the Curve

To ensure that a point lies on the curve, substitute the coordinates into the original equation:

$$3^2(4) 25$$ $$4^4 - 2(3) 256 - 6 250$$

Since these values are not equal, the point (3, 4) does not lie on the curve.

Conclusion

When calculating the slope of a curve at a specific point, always ensure that the point lies on the curve. If it does not, attempting to find the slope at that point is not only meaningless but also incorrect. Understanding this concept and using implicit differentiation correctly will help you navigate such problems.