91Ó°ÊÓ

Implicit differentiation is a method to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, we have the equation of the curve given by:
\[ y^2 = x^3 \]
The equation is not solved for y in terms of x or vice versa. To differentiate it, we treat y as a function of x and apply the chain rule.

Let's differentiate both sides with respect to x:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(x^3) \]
This gives:
\[ 2y \frac{dy}{dx} = 3x^2 \]
Here, we applied the chain rule on y^2, expressing y as a function of x, following from: \[ \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \]
This allows us to solve for the first derivative: \[ \frac{dy}{dx} = \frac{3x^2}{2y} \]
Implicit differentiation is handy when dealing with complex curves where isolating y is difficult.

To understand curvature, it's essential to find the second derivative of a function, which measures the curvature and concavity of the graph. Starting from the first derivative obtained, \[ \frac{dy}{dx} = \frac{3x^2}{2y} \], we differentiate again using the chain rule and implicit differentiation:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \bigg( \frac{3x^2}{2y} \bigg) \]
This involves several steps:

• Differentiate the numerator and denominator separately.
• Apply the chain rule for terms involving y, considering y = f(x).

After applying these rules, we get:
\[ \frac{d^2y}{dx^2} = \frac{ -3x ( \frac{dy}{dx} )}{2y} + \frac{3x^2}{2y^2} \bigg( \frac{dy}{dx} \bigg) \]
To simplify, we plug in the value of the first derivative at the given point:
\[ \frac{d^2y}{dx^2} \bigg|_{(\frac{1}{4}, \frac{1}{8})} = \frac{ -3 (\frac{1}{4}) ( \frac{3}{4} )}{2 (\frac{1}{8})} + \frac{3 ( \frac{1}{4}^2 ) (\frac{3}{4})}{2 ( \frac{1}{8}^2 )} \]
Simplifying this gives us the essential ingredients to calculate curvature.

The radius of curvature indicates how sharply a curve bends at a particular point. To find this, we first need to determine the curvature at the given point using the formula:
\[ K = \frac{ \big| \frac{d^2y}{dx^2} \big|}{\big( 1 + \big( \frac{dy}{dx} \big)^2 \big)^{3/2}} \]
Plugging in the values for the first and second derivatives we previously calculated, we find K.

The radius of curvature, \[ \rho \], is the reciprocal of the curvature:
\[ \rho = \frac{1}{K} \]
This concept is crucial in understanding the way a curve behaves near a particular point. For precise applications, like in engineering or physics, knowing the exact radius of curvature helps in designing paths and structures.

• Curvature tells us how much a curve deviates from being a straight line.
• The radius of curvature provides a quantitative measure of this deviation.