119. If $f$ is a differentiable function, $g(x) = f\!\left(\sqrt{1 + \tan^2 x}\right)$, and $g'\!\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}$, what is the value of $f'(2)$?
(1) $-\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $\dfrac{1}{2}$ (4) $1$

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, find $f ^ { \prime } ( x )$.

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) $2$