Higher Derivatives

Repeated differentiation produces higher derivatives. The following notations are equivalent.

First derivative

\[\mathrm{f}'(x)=\mathrm{f}^{(1)}(x)=\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)\]

Second derivative

\[\mathrm{f}''(x)=\mathrm{f}^{(2)}(x)=\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)\]

Third derivative

\[\mathrm{f}'''(x)=\mathrm{f}^{(3)}(x)=\frac{\mathrm{d}^3y}{\mathrm{d}x^3}=\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)\]

\(n\)th derivative

\[\mathrm{f}^{(n)}(x)=\frac{\mathrm{d}^ny}{\mathrm{d}x^n}\]

Take note

\[\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\ne\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2\]

\[\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)=\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2\]

The second derivative means differentiate the first derivative once more.