For each of the following functions use the rules for differentiability to prove that $f$ is differentiable. You may assume that the identity, constant, exponential and sine and cosine functions are differentiable on R:
(a) $f(x)=\mathrm{c}^{x} \cos x+1$
(b) $f(x)=\begin{gathered}1 \\ 1+x^{4}\end{gathered}$
(c) $f(x)=\frac{1+x^{2}}{1+x^{4}}$
(d) $f(x)-\tan ^{3} x \quad\left(x \neq n+\frac{1}{2} \pi\right)$
(e) $f(x)=\left\{\begin{array}{cl}x^{2} \cos \left(\begin{array}{c}1 \\ x\end{array}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.$
(f) $f(x)=x^{1 / 5} \quad(x>0)$
(g) $f(x)=\sinh ^{1} x \quad($ Hint $:$ The bijective function sinh: $\mathbb{R} \rightarrow[8$ is defined by sinh $x=\frac{1}{2}\left(\mathrm{c}^{x}-\mathrm{e}^{-x}\right)$.)