Class 12 Maths Continuity Differentiability Second Order Derivative

Second Order Derivative

Let y = f(x)

Differentiate w.r.t. x, we get

dy/dx = f ′(x)      ……….1

If f ′(x) is differentiable, we may again differentiate equation 1 w.r.t. x.

     d[dy/dx]/dx = d[f ′(x)]/dx

=> d2y/dx2 = f’’(x)   ……….2

Here, the left hand side d[dy/dx]/dx is called the second order derivative of y w.r.t. x and is denoted by

d2y/dx2. The second order derivative of f(x) is denoted by f ″(x). It is also denoted by D2 y or y″ or y2.

Problem: Find the second order derivatives of the given function: x2 + 3x + 2

Solution:

Let y = x2 + 3x + 2

Differentiate w.r.t. t, we get

      dy/dx = d(x2 + 3x + 2)/dx

=> dy/dx = d(x2)/dx + d(3x)/dx + d(2)/dx

=> dy/dx = 2x + 3 + 0

=> dy/dx = 2x + 3

Again, differentiate w.r.t. t, we get

      d2y/dx2 = d(2x + 3)/dx

=> d2y/dx2 = d(2x)/dx + d(3)/dx

=> d2y/dx2 = 2 + 0

=> d2y/dx2 = 2

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