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 $f(x)=x^{5}$ and $g(t)=e^{-5t}$

$\begin{array}{lll}f(x)=x^{5} &\implies f'(x)=5x^{4} &\implies f''(x)=5\cdot 4 x^{3}=20x^{3} \\ g(t)=e^{-5t} &\implies g'(t)=-5e^{-5t} &\implies g''(t)=-5\cdot -5e^{-5t}=25e^{-5t} \end{array}$

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 $f(t)=e^{2t}$ .

$\begin{aligned} f(t)&=e^{2t}\\ \\ f'(t) &= f^{(1)}(t)=2e^{2t}\\ \\ f''(t) &= f^{(2)}(t)=2^2e^{2t}=4e^{2t}\\ \\ f'''(t) &= f^{(3)}(t)=2^3e^{2t}=8e^{2t}\\ \\ f''''(t) &= f^{(4)}(t)=2^4e^{2t}=16e^{2t}\end{aligned}$
By pattern recognition we find the $n$ th derivative of $f(t)$ :

$f^{(n)}(t)=\frac{\dd ^nf}{\dd t^n}(t)=2^ne^{2t}$

New example