9. Second Order Derivative

Second Order Derivative:

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.

Example: Find d²y/dx², if y = x³?

Solution:

Given that, y = x³

Then, first derivative will be
dy/dx = d/dx (x³) = 3x²  
 

Again, we will differentiate further to find its 
second derivative,
 

Therefore, d²y/dx² = d/dx (dy/dx)

                              = d/dx (3x²)

                              = 6x                

Example : Find d²y/dx², if y = Asinx + Bcosx, Where A and B are constants? 

Solution:

Given that, y = Asinx + Bcosx

Then, first derivative will be

dy/dx = d/dx (Asinx + Bcosx)

          = A d/dx (sinx) + B d/dx (cosx)

          = A(cosx) + B(-sinx)

          = Acosx – Bsinx

Again, we will differentiate further to find its second derivative,

d²y/dx²  = d/dx (dy/dx)

             = d/dx (Acosx – Bsinx)

             = A d/dx (cosx) – B d/dx (sinx)

             = A(-sinx) – B(cosx)

             = -Asinx – Bcosx

             = -(Asinx + Bcosx)

             = -y

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