Let for a differentiable function $f : ( 0 , \infty ) \rightarrow R , f ( x ) - f ( y ) \geq \log _ { e } \left( \frac { x } { y } \right) + x - y , \forall x , y \in ( 0 , \infty )$. Then $\sum _ { n = 1 } ^ { 20 } f ^ { \prime } \left( \frac { 1 } { n ^ { 2 } } \right)$ is equal to

We are to differentiate
$$f ( x ) = \int _ { 0 } ^ { 2 x } \left( t ^ { 2 } - x ^ { 2 } \right) \sin 3 t \, d t$$
with respect to $x$.
(1) We know that if $g ( t )$ is a continuous function and $G ( t )$ is one of its primitive functions, then
$$\int _ { 0 } ^ { 2 x } g ( t ) d t = G ( 2 x ) - G ( 0 )$$
By differentiating both sides of this equality with respect to $x$, we have
$$\frac { d } { d x } \int _ { 0 } ^ { 2 x } g ( t ) d t = \mathbf { A }$$
where $\mathbf{A}$ is the appropriate expression from among the following (0) $\sim$ (7).
(0) $g ( x )$
(1) $\frac { 1 } { 2 } g ( x )$
(2) $2 g ( x )$
(3) $g ( 2 x )$
(4) $\frac { 1 } { 2 } g ( 2 x )$
(5) $2 g ( 2 x )$ (6) $g ( x ) - g ( 0 )$ (7) $g ( 2 x ) - g ( 0 )$
(2) We know that $f ( x ) = \int _ { 0 } ^ { 2 x } t ^ { 2 } \sin 3 t \, d t - \int _ { 0 } ^ { 2 x } x ^ { 2 } \sin 3 t \, d t$.
Since
$$\frac { d } { d x } \int _ { 0 } ^ { 2 x } t ^ { 2 } \sin 3 t \, d t = \mathbf { B } x ^ { 2 } \sin \mathbf { C } x$$
and
$$\frac { d } { d x } \int _ { 0 } ^ { 2 x } x ^ { 2 } \sin 3 t \, d t = \frac { \mathbf { D } } { \mathbf { E } } x ( - \cos \mathbf { F } x + \mathbf { G } + \mathbf { H } x \sin \mathbf { I } x )$$
we obtain
$$f ^ { \prime } ( x ) = \frac { \mathbf { D } } { \mathbf { E } } x ( \cos \mathbf { J } x - \mathbf { K } + \mathbf { L } x \sin \mathbf { M } x )$$

Let $F ( x )$ be a polynomial with real coefficients and $F ^ { \prime } ( x ) = f ( x )$ . It is known that $f ^ { \prime } ( x ) > x ^ { 2 } + 1.1$ holds for all real numbers $x$. Select the correct options.
(1) $f ^ { \prime } ( x )$ is an increasing function
(2) $f ( x )$ is an increasing function
(3) $F ( x )$ is an increasing function
(4) $[ f ( x ) ] ^ { 2 }$ is an increasing function
(5) $f ( f ( x ) )$ is an increasing function

On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. Which of the following is the derivative of $f(x)$? (Single choice, 2 points)
(1) $x^{2} - 9x + 15$
(2) $3x^{3} - 18x^{2} + 15x - 4$
(3) $3x^{3} - 18x^{2} + 15x$
(4) $3x^{2} - 18x + 15$
(5) $x^{2} - 18x + 15$

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 $f ^ { \prime } ( x )$.

$$f ( x ) = \sin ^ { 2 } \left( 3 x ^ { 2 } + 2 x + 1 \right)$$
Given this, what is the value of $f ^ { \prime } ( 0 )$?
A) $2 \cos 2$
B) $2 \cos 3$
C) $6 \sin 1$
D) $4 \sin 2$
E) $2 \sin 2$

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) $2 e ^ { 2 }$

A function f is defined on a subset of the set of real numbers as
$$f ( x ) = \frac { x ^ { 2 } - 4 x + 4 } { x - 2 } + \frac { x ^ { 2 } - 6 x + 9 } { 2 x - 6 }$$
Accordingly, $$\lim _ { x \rightarrow 2 } f ( x ) + \lim _ { x \rightarrow 3 } f ( x )$$
what is the value of this expression?
A) $\frac { 3 } { 2 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 4 }$

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

A function f is defined on the set of real numbers as
$$f ( x ) = x ^ { 2 } + x - 4$$
A function g defined and continuous on the set of real numbers has a derivative $g ^ { \prime }$ such that $g ^ { \prime } ( x ) = 0$ only for $x = 2$. Accordingly, the product of the x values satisfying
$$( g \circ f ) ^ { \prime } ( x ) = 0$$
is what?
A) 0
B) 1
C) 3
D) 4
E) 6

$$\lim_{x \rightarrow 1} \frac{(1 - \sqrt{x}) \cdot (\sqrt[3]{x} - 2)}{-x^{2} + 9x - 8}$$
What is the value of this limit?
A) 1 B) $\dfrac{1}{2}$ C) $\dfrac{1}{7}$ D) $\dfrac{1}{14}$ E) $\dfrac{1}{18}$

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