Higher derivatives

Differentiation, derivatives and Taylor approximations: Higher-Order derivatives

Higher derivatives

Differentiation of a function \(f(x)\) yields the derivative \(f'(x)\), which is also written as \(\displaystyle\frac{\dd f}{\dd x}(x)\) and \(\displaystyle\frac{\dd}{\dd x}f(x)\tiny.\) This derivative is a function of \(x\) that we can differentiate again (at least for very smooth functions). This yields the second derivative of \(f(x)\). Common formats for this are \(\displaystyle f''(x), \frac{\dd^2f}{\dd x^2}\!(x)\) and \(\displaystyle\frac{\dd^2}{\dd x^2}\!f(x)\)
(Note the different placement of the number 2 above and below the 'division' line in the last two notations).

Calculate the second derivative of \(f(x)=x^{9}\) and \(g(t)=e^{-2t}\)

\[\begin{array}{lll}f(x)=x^{9} &\implies f'(x)=9x^{8} &\implies f''(x)=9\cdot 8 x^{7}=72x^{7} \\
g(t)=e^{-2t} &\implies g'(t)=-2e^{-2t} &\implies g''(t)=-2\cdot -2e^{-2t}=4e^{-2t} \end{array}\]

New example

We have formed the derivative of a derivative formed and we can go. At \(n\) time differentiating feature \(f(x)\) we get the \(n\)-th derivative. In general, the \(n\)-th derivative with \(n>2\) is written in one of the following formats: \(\displaystyle f^{(n)}(x), \frac{\dd^nf}{\dd x^n}(x)\) and \(\displaystyle\frac{\dd^n}{\dd x^n}\!f(x)\).

Calculate the first, second, third, and fourth derivative of \(f(t)=\ln(t)\).

\[\begin{aligned} f(t)&=\ln(t)\\ \\ f'(t) &= f^{(1)}(t)=\frac{1}{t}=t^{-1}\\ \\ f''(t) &= f^{(2)}(t)=-t^{-2}=-\frac{1}{t^2}\\ \\ f'''(t) &= f^{(3)}(t)=2t^{-3}=\frac{2}{t^3}\\ \\ f''''(t) &= f^{(4)}(t)=-2\cdot 3t^{-4}=-\frac{6}{t^4}\end{aligned}\]
By pattern recognition we find the \(n\)th derivative of \(f(t)\): \[f^{(n)}(t)=\frac{\dd ^nf}{\dd t^n}(t)=(-1)^{n-1}(n-1)!\,t^{-n}\] Hierbij is \((n-1)!\) gedefinieerd als \(1 \times 2\times 3\times\cdots\times (n-2)\times(n-1)\)

New example