9.2 Second Derivatives of Parametric Equations

If we wanted to find the second derivative of a parametric function d^2y/dx^2, we would simply use the chain rule: ⛓️

Here's a more in-depth description of the formula above:

Finding the second derivative of a parametric function involves taking the derivative of the first derivative of the function. In order to do this, we first need to find the first derivative of the parametric function, which is known as the parametric derivative as seen in the previous section. 🛒

As a reminder, the parametric derivative is found by taking the ratio of the derivative of the y-coordinate function with respect to the parameter (dy/dt) to the derivative of the x-coordinate function with respect to the parameter (dx/dt). This is the equation dy/dx = dy/dt / dx/dt.

Once we have found the parametric derivative, we can then take the derivative of this equation with respect to the parameter to find the second derivative. This is done by taking the derivative of both sides of the equation with respect to the parameter.

On the left side, the second derivative is denoted as d^2y/dx^2. On the right side, the second derivative is found by taking the derivative of the ratio dy/dt / dx/dt with respect to the parameter. This can be done using the quotient rule for derivatives, which states that the derivative of a ratio is equal to the difference of the product of the derivative of the numerator and the denominator, and the product of the numerator and the derivative of the denominator, all divided by the square of the denominator. 🤣

Example: Show that the cycloid defined by x(t) = 2(t − sint) and y(t)= 2(1 − cost) is concave down on t E (0, 2)

💡 Remember: Concavity is determined by the second derivative 

(1) Consider the parametric equations x = t^3 - 3t and y = t^2 + 2t - 5. Find the second derivative of y with respect to x.

(2) The parametric equation of a curve is given by x = cos^3(t) and y = sin^3(t). At the point (1/2,1/2), what is the value of the second derivative of y with respect to x?

(3) A particle moves along the curve defined by the parametric equations x = t^2 - 2t and y = t^3 - 3t^2 + 4t. At the point (-1,3), what is the value of the second derivative of y with respect to x?

(4) A curve is defined by the parametric equations x = sin(t) and y = cos(t). The second derivative of y with respect to x can be expressed as...?

(5) A car is moving along a curved road that can be represented by the parametric equations x = t^3 and y = t^4. At the point (2,8), what is the value of the second derivative of y with respect to x?