Second Order Derivatives

 The second order derivative of a function y = f(x) is the derivative of the first derivative of the function.

The first derivative of the function f is dy/dx.

The derivative of the first derivative is :d/dx[dy/dx] = d²y/dx²

The relation equality being commutative, we can write: d²y/dx² = d/dx[dy/dx].

Let's apply the formula of the first derivative. According to this formula, the derivative of y = f(x) is the derivative of the function y with respect to t divided by the derivative of x with respect to t.

The function here is dy/dx. Let's calculate its derivative:

 d²y/dx² = d/dt[dy/dx]./dx/dt.

The second order derivative is the derivative of the derivative of the first derivative with respect to t divided by the derivative of x with respect to t.

Examples

Calculate the second derivative  d²y/dx² for the plane curve defined by the parametric equations:

 Solution

We have :

 d²y/dx² = d/dt[dy/dx]./dx/dt.

Let's calculate dy/dx. According to the formula of the derivative, we know that the derivative is equal to the derivative of y with respect to t divided by the derivative of x with respect to t.

dy/dx = y'(t)/x'(t) = 2/2t = 1/t

Let's calculate dx/dt also:

dx/dt = 2t

The expression of d²y/dx² becomes:

d²y/dx² = d/dt (1/t)/2t = -1/t²/2t = (-1/t²)(1/2t) = -1/2t³

Practice

Calculate the second derivative  d²y/dx² for the plane curve defined by the parametric equations:

Locate any critical points on its curve.