21.19 - Second Order Derivative

In the previous section, we completed a discussion on parametric functions and their differentiation. In this section, we will see second order derivative.

The basic details can be written in 4 steps:
1. Consider the function y = f(x)
• If f(x) is differentiable, then we can write:
$\rm{\frac{dy}{dx}}\,=\,f'(x)$
2. We know that, f'(x) is also a function. If this f'(x) is differentiable, we can find it’s derivative also. In such a situation, we write:
$\rm{\frac{d}{dx} \left(\frac{dy}{dx} \right)\,=\,f''(x)}$
• $\rm{f''(x)}$ is known as the second order derivative of f(x).
• $\rm{\frac{d}{dx} \left(\frac{dy}{dx} \right)}$ is denoted as $\rm{\frac{d^2 y}{dx^2}}$
• So we get: $\rm{\frac{d^2 y}{dx^2}\,=\,f''(x)}$
3. If y = f(x) then we can write in the following ways also:
• $\rm{\frac{d^2 y}{dx^2}\,=\,f''(x)\,=\,D^2 y}$
• $\rm{\frac{d^2 y}{dx^2}\,=\,f''(x)\,=\,y''}$
• $\rm{\frac{d^2 y}{dx^2}\,=\,f''(x)\,=\,y_2}$
4. We can write steps similar to the above three, to define third order derivative, fourth order derivative, fifth order derivative, so on . . .

Let us see some solved examples

Solved example 21.60
$\rm{\text{Find}~\frac{d^2 y}{dx^2}~\text{if}~y = x^3 + tan x}$.
Solution:

 

Solved example 21.61
If y = A sin x + B cos x, then prove that $\rm{\frac{d^2 y}{dx^2}~+~y = 0}$.
Solution:

Solved example 21.62
If y = 3e^2x + 2e^2x, then prove that $\rm{\frac{d^2 y}{dx^2}~-~5\frac{dy}{dx}~+~6y = 0}$.
Solution:

Solved example 21.63
If y = sin⁻¹x, show that $\rm{(1-x^2) \frac{d^2 y}{dx^2}~-~x \frac{dy}{dx} = 0}$.
Solution:

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The link below gives a few more solved examples:

Exercise 21.7

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In the next section, we will see mean value theorem.

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