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3.33 Acceleration as the second derivative

So, taking the limit as $h\rightarrow 0$ ,

f^{\prime}(x)=\lim_{h\rightarrow 0}{{f(x+h)-f(x)}\over{h}}=({\hbox{velocity at time}}\quad x);

f^{\prime\prime}(x)=({\hbox{acceleration at time}}\quad x).

Example

To find the higher order derivatives of $y=x^{n}$ .

Solution. We compute the derivatives in a column to help us recognise patterns: