In the open interval $(1, e)$, I. The function $\sin ( \ln ( x ) )$ is increasing. II. The function $\cos ( \ln ( x ) )$ is increasing. III. The function $\tan ( \ln ( \mathrm { x } ) )$ is increasing. Which of these statements are true?
A) Only I
B) Only II
C) I and III
D) II and III
E) I, II and III

Let $f ( x ) = e ^ { x }$. The function $g$ is defined as
$$g ( x ) = ( f \circ f ) ( x )$$
Accordingly, what is the value of the derivative of the $\mathbf { g }$ function at the point $\mathbf { x } = \boldsymbol { \ln } \mathbf { 2 }$, that is, $\mathbf { g } ^ { \prime } ( \ln 2 )$?
A) e
B) $\ln 2$
C) $2 \ln 2$
D) $e ^ { 2 }$
E) $2 e ^ { 2 }$

$\lim _ { x \rightarrow \pi } \frac { x ^ { 2 } \cdot \sin ( \pi - x ) + \pi ^ { 2 } \cdot \sin ( x - \pi ) } { ( x - \pi ) ^ { 2 } }$\ What is the value of this limit?\ A) $- 2 \pi$\ B) $- \pi$\ C) $\pi$\ D) $2 \pi$\ E) $3 \pi$

Let k be a real number. For differentiable functions f and g defined on subsets of the set of real numbers,
$$f ( x ) = g \left( x ^ { 2 } \right) + k x ^ { 3 }$$
equality is satisfied.
Given that $$f ^ { \prime } ( - 1 ) = g ^ { \prime } ( 1 ) = 2$$
what is k?
A) 2
B) 1
C) 0
D) - 1
E) - 2

Let $n$ be a positive integer and $a$ be a non-zero real number. For a polynomial function $f$ with degree $n$ and leading coefficient $a$,
$$\left((f(x))^{3}\right)' = \left(f'(x)\right)^{4}$$
is satisfied.
Accordingly, what is the product $a \cdot n$?
A) $\frac{1}{4}$ B) $\frac{1}{3}$ C) $\frac{1}{2}$ D) $1$ E) $2$

In the rectangular coordinate plane, for a function $y \geq f(x)$,

• the tangent line at the point $(2, f(2))$ is $y = 3x - 1$
• the tangent line at the point $(5, f(5))$ is $y = 2x + 4$

Accordingly, for the function $$g(x) = x^{2} \cdot (f \circ f)(x)$$
what is the value of $g'(2)$?
A) 64 B) 72 C) 80 D) 88 E) 96