If $y = x \sin x$, then $\frac { d y } { d x } =$
(A) $\sin x + \cos x$
(B) $\sin x + x \cos x$
(C) $\sin x - x \cos x$
(D) $x ( \sin x + \cos x )$
(E) $x ( \sin x - \cos x )$

Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = x ^ { 4 }$. On what intervals is the rate of change of $f ( x )$ greater than the rate of change of $g ( x )$ ?
(A) $( 0.831, 7.384 )$ only
(B) $( - \infty , 0.831 )$ and $( 7.384 , \infty )$
(C) $( - \infty , - 0.816 )$ and $( 1.430, 8.613 )$
(D) $( - 0.816, 1.430 )$ and $( 8.613 , \infty )$
(E) $( - \infty , \infty )$

For the function $f ( x ) = x ^ { 3 } + 7 x + 3$, what is the value of $f ^ { \prime } ( 1 )$? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12

For the function $f ( x ) = 2 x ^ { 3 } + x + 1$, find the value of $f ^ { \prime } ( 1 )$. [3 points]

For the function $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + x - 1$, what is the value of $f ^ { \prime } ( 1 )$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

11. Given the function $f ( x ) = a x \ln x , x \in ( 0 , + \infty )$, where $a$ is a real number, and $f ^ { \prime } ( x )$ is the derivative of $f ( x )$. If $f ^ { \prime } ( 1 ) = 3$, then the value of $a$ is $\_\_\_\_$.

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

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) $x^{2} - 18x + 15$