# Higher Order Derivatives – Definition and Example

Here you will learn higher order derivatives of functions with examples.

Let’s begin –

## Higher Order Derivatives Definition and Notations

If y = f(x), then $$dy\over dx$$, the derivative of y with respect to x, is itself, in general, a function of x and can be differentiated gain.

To fix up the idea, we shall call $$dy\over dx$$ as the first order derivative of y with respect to x and the derivative of $$dy\over dx$$ with respect to x as the second order derivative of y with respect to x and will be denoted by $$d^2y\over dx^2$$.

Similarly the derivative of $$d^2y\over dx^2$$ with respect to x will be termed as the third order derivative of y with respect to x and will be denoted by $$d^3y\over dx^3$$ and so on. The $$n^{th}$$ order derivative of y with respect to x will be denoted by $$d^ny\over dx^n$$.

If y = f(x), then the other alternative notations for

$$dy\over dx$$, $$d^2y\over dx^2$$, $$d^3y\over dx^3$$, …… , $$d^ny\over dx^n$$ are

$$y_1$$, $$y_2$$, $$y^3$$, …… , $$y_n$$

y’ , y” , y”’ , ……. , $$y^(n)$$

Dy, $$D^2$$y , $$D^3$$y , ….. , $$D^n$$y

f'(x) , f”(x) , f”'(x) , …… , $$f^{n}$$ (x)

The value of these derivatives at x = a are denoted by $$y_n$$ (a), $$y^n$$ (a) , $$D^n$$y (a) or, $$({d^ny\over dx^n})_{x = a}$$.

Example : If y = $$sin^{-1}x$$, show that $$d^2y\over dx^2$$ = $$x\over {(1 – x^2)^{3/2}}$$

Solution : We have, y = $$sin^{-1}x$$.

On differentiating with respect to x, we get

$$dy\over dx$$ = $$1\over \sqrt{1 – x^2}$$

On differentiating again with respect to x, we get

$$d^2y\over dx^2$$ = $$d\over dx$$$$({1\over \sqrt{1 – x^2}})$$

= $$d\over dx$$$$[{(1 – x^2)^{-1/2}}]$$

= $$-1\over 2$$ $$(1 – x^2)^{-3/2}$$ $$\times$$ $$d\over dx$$ ($$1 – x^2$$)

$$\implies$$ $$d^2y\over dx^2$$ = -$$1\over 2(1 – x^2)^{3/2}$$(-2x) = $$x\over {(1 – x^2)^{3/2}}$$