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) = \(x\over {(1 – x^2)^{3/2}}\)

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