Question

Assume a curve is given by the parametric equations \(x=g(t)\) and \(y=h(t),\) where \(g\) and \(h\) are twice differentiable. Use the Chain Rule to show that $$y^{\prime \prime}(x)=\frac{x^{\prime}(t) y^{\prime \prime}(t)-y^{\prime}(t) x^{\prime \prime}(t)}{\left(x^{\prime}(t)\right)^{3}}$$

Short answer

Question: Given the parametric equations \(x = g(t)\) and \(y = h(t)\), with g and h being twice differentiable, find the second derivative of y with respect to x, denoted as \(y^{\prime\prime}(x)\). Answer: To find the second derivative of y with respect to x, we can follow these steps: 1. Find the first derivatives of x and y with respect to t: \(x'(t)\) and \(y'(t)\). 2. Find the second derivatives of x and y with respect to t: \(x''(t)\) and \(y''(t)\). 3. Compute the derivative of \(y'(t)\) with respect to \(x'(t)\) using the Chain Rule: \(\frac{dy'(t)}{dx'(t)} = \frac{y''(t)}{x''(t)}\). 4. Multiply both sides by \(x'(t)\) to isolate y''(t): \(y''(t) x'(t) = \frac{y''(t)}{x''(t)}x'(t)^2\). 5. Divide both sides by \(x'(t)^2\) to find the desired result, \(y''(x)\): \(y''(x) = \frac{x'(t) y''(t) - y'(t) x''(t)}{x'(t)^3}\).

Step-by-step solution

Find the first derivatives of x and y with respect to t

We will begin by finding the first derivatives of x and y with respect to t: $$x'(t) = \frac{d}{dt}g(t) \quad \text{and} \quad y'(t) = \frac{d}{dt}h(t)$$

Find the second derivatives of x and y with respect to t

Next, we find the second derivatives of x and y with respect to t: $$x''(t) = \frac{d^2}{dt^2}g(t) \quad \text{and} \quad y''(t) = \frac{d^2}{dt^2}h(t)$$

Find the derivative of y'(t) with respect to x'(t)

Now, we will compute the derivative of \(y'(t)\) with respect to \(x'(t)\) using the Chain Rule: $$\frac{dy'(t)}{dx'(t)} = \frac{y''(t)}{x''(t)}$$

Solve for y''(x) using the relationship between dx and dt

Multiply both sides by \(x'(t)\) to isolate y''(t): $$y''(t) x'(t) = \frac{y''(t)}{x''(t)}x'(t)^2$$

Divide both sides by x'(t)^2

Lastly, we divide both sides by \(x'^3(t)\) to find the desired result, \(y''(x)\): $$y''(x) = \frac{x'(t) y''(t) - y'(t) x''(t)}{x'(t)^3}$$

Key concepts

Parametric Equations

Parametric equations are a way to define a curve using a parameter, usually denoted as \( t \). Rather than expressing \( y \) as a direct function of \( x \), both \( x \) and \( y \) are expressed in terms of \( t \).
For example, the parametric equations \( x = g(t) \) and \( y = h(t) \) describe a curve by tracing the coordinates \( (x,y) \) as \( t \) varies. This technique is especially useful when dealing with curves that do not have a single or straightforward function form, such as circles or other complex shapes.
By having two equations, we can capture a larger range of behaviors and interactions between \( x \) and \( y \), allowing us to explore curves with cusps, loops, and crossings more easily. Parametric equations offer flexibility in how we represent the path or trajectory of an object, making it crucial for modeling situations in physics and engineering.

Derivatives

Derivatives are fundamental in calculus and represent the rate at which one quantity changes with respect to another. When dealing with parametric equations, we often need to find derivatives with respect to the parameter \( t \).
For the parametric curve defined by \( x = g(t) \) and \( y = h(t) \), the first derivatives \( x'(t) \) and \( y'(t) \) indicate how \( x \) and \( y \) change with increasing \( t \). These derivatives are critical for understanding the curve's behavior, such as direction and speed of change.
To find the derivative of \( y \) with respect to \( x \), we use the relationship \( \frac{dy}{dx} = \frac{y'(t)}{x'(t)} \), often called the parametric derivative. This helps us translate changes in \( t \) into changes in \( x \) and \( y \), aligning the parametric adjustments with the Cartesian plane.
Second derivatives, such as \( y''(t) \) and \( x''(t) \), give information about the acceleration or concavity of the curve, and can be derived by differentiating the first derivatives. Understanding these derivatives is key to analyzing the nature and shape of complex paths.

Differentiability

Differentiability is a condition that ensures a function has a well-defined tangent line at every point in its domain. It is a crucial concept for both practical applications and theoretical analysis.
For parametric equations, differentiability means that both \( g(t) \) and \( h(t) \) must be differentiable, thus allowing us to compute their derivatives. The assumption that these functions are twice differentiable ensures their derivatives, \( x'(t) \) and \( y'(t) \), are themselves smooth and continuous.
This property is what allows us to reliably apply tools like the Chain Rule. The Chain Rule is essential when dealing with compositions of multiple functions, like when finding the derivative of a parametric equation. It uses the derivatives of the component functions to find the overall derivative.
In the context of curves described by parametric equations, differentiability ensures that changes in \( t \) translate smoothly and predictably into changes in \( x \) and \( y \), enabling accurate modeling and analysis of complex systems or motions.