Excercise 1
Find the derivatives of the following functions :
\((a) f (x) =(x + \frac{1}{x})^{2} , x \in \mathbb{R}, x\neq0,\)
\((b) f (x) = \sum_{k=0}^{10} 2^{k}x^{k}, x \in \mathbb{R},\)
\((c) f (x) = x^{(a^{x})}, a > 0, x \in (0, \infty).\)
Solution
\((a) f (x) = x^{2} + 2 + \frac{1}{x^{2}} \Rightarrow f'(x)=2x-\frac{2}{x^{3}}\)
\((b) f'(x)=(\sum_{k=0}^{10}2^{k}x^{k})'= \sum_{k=1}^{10}2^{k}kx^{k-1}\)
(c) We have \((ax)'= a^{x}\ln a\). Also:
\((x^{a^{x}})'=(e^{a^{x}\ln x})'= x^{a^{x}\ln x}( a^{x}\ln a \ln x + \frac{a^{x}}{x})=x^{a^{x}} a^{x} (\ln a \ln x + \frac{1}{x})\).