If y = f(x), then dy/dx the derivative of y with respect to x, is itself, in general or function of x and can be differentiated again.
To fix up the idea, we shall call dy/dx as the first order derivative of y with respect to x and the derivative of dy/dx w.r.t. x as the second order derivative of y w.r.t x and will be denoted by d²y/dx², similarly the derivative of d²y/dx² w.r.t. x will be termed as the third order derivative of y w.r.t. x and will be denoted by d³y/dx³ and soon.
The nthorder derivative of y w.r.t x will be denoted by dⁿy/dxⁿ.
To fix up the idea, we shall call dy/dx as the first order derivative of y with respect to x and the derivative of dy/dx w.r.t. x as the second order derivative of y w.r.t x and will be denoted by d²y/dx², similarly the derivative of d²y/dx² w.r.t. x will be termed as the third order derivative of y w.r.t. x and will be denoted by d³y/dx³ and soon.
The nthorder derivative of y w.r.t x will be denoted by dⁿy/dxⁿ.
If
y = f(x) then the other alternative notation for dy/dx, d²y/dx², d³y/dx³, …, dⁿy/dxⁿ are y₁, y₂, y₃, … yn
y’, y’’, y’’’, … y(n)
Dy, Dy², D³y, …, Dⁿy
f’(x), f’’(x), f’’’(x), … fⁿ(x)
The values of these derivatives at are denoted by yn
(a), yⁿ (a), Dⁿy (a), fⁿ (a) or
.
.
Examples:
If x = sinθ, y = cospθ prove that (1 -
x²) y₂ - xy₁ + p²y = 0, where y₂ = d²y/dx² and y₁ = dy/dx.
Solution: We have,
x = sinθ and y = cospθ
Differentiating
both sides w.r.t. x we get
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