Michael J.

Each of Exercises $1-4$ gives a value of sinh $x$ or cosh $x$ . Use the definitions and the identity cosh $^{2} x-\sinh ^{2} x=1$ to find the values of the remaining five hyperbolic functions.
$$\sin x=-\frac{3}{4}$$

Prove the identities
$\begin{aligned} \sinh (x+y) &=\sinh x \cosh y+\cosh x \sinh y \\ \cosh (x+y) &=\cosh x \cosh y+\sinh x \sinh y \end{aligned}$
Then use them to show that
a. \sinh 2 x=2 \sinh x \cosh x
b. \cosh 2 x=\cosh ^{2} x+\sinh ^{2} x

Which of the following functions grow faster than $e^{x}$ as $x \rightarrow \infty ?$ Which grow at the same rate as $e^{x} ?$ Which grow slower?
a. $x-3 \quad$ b. $x^{3}+\sin ^{2} x$
c. $\sqrt{x} \quad$ d. $4^{x}$
e. $(3 / 2)^{x} \quad$ f. $e^{x / 2}$
g. $e^{x} / 2 \quad$ h. $\log _{10} x$

Which of the following functions grow faster than $e^{x}$ as $x \rightarrow \infty ?$ Which grow at the same rate as $e^{x} ?$ Which grow slower?
a. $10 x^{4}+30 x+1 \quad$ b. $x \ln x-x$
c. $\sqrt{1+x^{4}}$ d. $(5 / 2)^{x}$
e. $e^{-x} \quad$ f. $x e^{x}$
g. $e^{\cos x}$ h. $e^{x-1}$

Which of the following functions grow faster than $x^{2}$ as $x \rightarrow \infty$ ? Which grow at the same rate as $x^{2}$ ? Which grow slower?
a. $x^{2}+4 x$ b. $x^{5}-x^{2}$
c. $\sqrt{x^{4}+x^{3}} \quad$ d. $(x+3)^{2}$
g. $x^{3} e^{-x}$ h. 8$x^{2}$

In Exercises $13-24,$ find the derivative of $y$ with respect to the appropriate variable.
$$y=\left(x^{2}+1\right) \operatorname{sech}(\ln x)$$