Second-Order Derivatives of a Function in Parametric Form
To calculate the second derivative of the function in the parametric form we use the chain rule twice. Hence to find the second derivative, we find the derivative with respect to t of the first derivative and then divide by the derivative of x with respect to t. Suppose that x = x(t) and y = y(t), then its Parametric form in Second Order:
First Derivative: dy/dx = (dy/dt) / (dx/dt)
Second Derivative: d2y/dx2 = d/dx (dy/dx)
= d/dt (dy/dx) / (dx/dt)
Note: It is totally wrong to write the above formula as d2y/dx2 = (d2y/dt2) / (d2x/dt2)
Example: If x = t + cost, y = sint, find the second derivative.
Solution:
Given that, x = t + cost and y = sint
First Derivative,
dy/dx = (dy/dt) / (dx/dt)
= (d/dt (sint)) / (d/dt (t + cost))
= (cost) / (1 – sint) —- (1)
Second Derivative,
d2y / dx2 = d/dx (dy/dx)
= d/dx (cost / 1 – sint) —- (from eq.(1))
= d/dt (cost / 1 – sint) / (dx/dt) —- (chain rule)
= ((1 – sint) (-sint) – cost(-cost)) / (1 – sint)2 / (dx/dt) —- (quotient rule)
= (-sint + sin2t + cos2t) / (1 – sint)2 / (1 – sint)= (-sint + 1) / (1 – sint)3
= 1 / (1 – sint)2
Note:
1) Quotient Rule of Differentiation: dy/dx = v(du/dx) – u(dv/dx) / v2
2) Chain Rule: dy/dx = (dy/du) . (du/dx)
Second Order Derivatives: Rules , Formula and Examples (Class 12 Maths)
The Second Order Derivative is defined as the derivative of the first derivative of the given function. The first-order derivative at a given point gives us the information about the slope of the tangent at that point or the instantaneous rate of change of a function at that point.
Second-Order Derivative gives us the idea of the shape of the graph of a given function. The second derivative of a function f(x) is usually denoted as f”(x). It is also denoted by D2y or y2 or y” if y = f(x).
Let y = f(x)
Then, dy/dx = f'(x)
If f'(x) is differentiable, we may differentiate (1) again w.r.t x. Then, the left-hand side becomes d/dx(dy/dx) which is called the second order derivative of y w.r.t x.
In this article, we have covered Graphical Representation of Second-Order Derivatives along with formulas, examples and many more.
Table of Content
- Second Order Derivatives Overview
- Second Order Derivatives Examples
- Second-Order Derivatives of a Function in Parametric Form
- Graphical Representation of Second-Order Derivatives
- Concavity of Function
- Concave Down
- Points of Inflection