91Ó°ÊÓ

(a) Let \(h(x)=\sqrt{x}=x^{1 / 2}\) for \(x \geq 0 .\) Use the definition of derivative to prove that \(h^{\prime}(x)=\frac{1}{2} x^{-1 / 2}\) for \(x>0\) (b) Let \(f(x)=x^{1 / 3}\) for \(x \in \mathbb{R}\) and use the definition of derivative to prove that \(f^{\prime}(x)=\frac{1}{3} x^{-2 / 3}\) for \(x \neq 0\) (c) Is the function \(f\) in part (b) differentiable at \(x=0 ?\) Explain.

Show that \(e x \leq e^{x}\) for all \(x \in \mathbb{R}\)

Show that \(\sin x \leq x\) for all \(x \geq 0 .\) Hint: Show that \(f(x)=x-\sin x\) is increasing on \([0, \infty)\)

Prove Leibniz' rule $$(f g)^{(n)}(a)=\sum_{k=0}^{n}\left(\begin{array}{l}n \\\k\end{array}\right) f^{(k)}(a) g^{(n-k)}(a)$$ provided both \(f\) and \(g\) have \(n\) derivatives at \(a\). Here \(h^{(0)}\) signifies the jth derivative of \(h\) so that \(h^{(0)}=h, h^{(1)}=h^{\prime}, h^{(2)}=h^{\prime \prime},\) etc. Also, \(\left(\begin{array}{l}n \\ k\end{array}\right)\) is the binomial coefficient that appears in the binomial expansion; see Exercise \(1.12 .\) Hint: Use mathematical induction. For \(n=1,\) apply Theorem \(28.3(\text { iii })\)