$x$	$f ( x )$	$f ^ { \prime } ( x )$	$g ( x )$	$g ^ { \prime } ( x )$
1	2	- 4	- 5	3
2	- 3	1	8	4

The table above gives values of the differentiable functions $f$ and $g$ and their derivatives at selected values of $x$. If $h$ is the function defined by $h ( x ) = f ( x ) g ( x ) + 2 g ( x )$, then $h ^ { \prime } ( 1 ) =$
(A) 32
(B) 30
(C) - 6
(D) - 16

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 $f$ be the function defined for all real numbers $x$ in the interval $] 0 ; + \infty [$ by:
$$f ( x ) = \frac { \mathrm { e } ^ { 2 x } } { x }$$
The expression of the second derivative $f ^ { \prime \prime }$ of $f$ is given, defined on the interval $] 0 ; + \infty [$ by:
$$f ^ { \prime \prime } ( x ) = \frac { 2 \mathrm { e } ^ { 2 x } \left( 2 x ^ { 2 } - 2 x + 1 \right) } { x ^ { 3 } } .$$

1. The function $f ^ { \prime }$, the derivative of $f$, is defined on the interval $] 0 ; + \infty [$ by: a. $f ^ { \prime } ( x ) = 2 \mathrm { e } ^ { 2 x }$ b. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( x - 1 ) } { x ^ { 2 } }$ c. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 2 x - 1 ) } { x ^ { 2 } }$ d. $f ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { 2 x } ( 1 + 2 x ) } { x ^ { 2 } }$.
2. The function $f$: a. is decreasing on $] 0 ; + \infty [$ b. is monotonic on $] 0 ; + \infty [$ c. admits a minimum at $\frac { 1 } { 2 }$ d. admits a maximum at $\frac { 1 } { 2 }$.
3. The function $f$ has the following limit as $x \to + \infty$: a. $+ \infty$ b. 0 c. 1 d. $\mathrm { e } ^ { 2 x }$.
4. The function $f$: a. is concave on $] 0 ; + \infty [$ b. is convex on $] 0 ; + \infty [$ c. is concave on $] 0 ; \frac { 1 } { 2 } ]$ d. is represented by a curve admitting an inflection point.

Let $f$ be the function defined on $]0; +\infty[$ by $f(x) = x^2 \ln x$. The expression of the derivative function of $f$ is: a. $f'(x) = 2x \ln x$. b. $f'(x) = x(2\ln x + 1)$. c. $f'(x) = 2$. d. $f'(x) = x$.

For a polynomial function $f ( x )$, define the function $g ( x )$ as $$g ( x ) = x ^ { 2 } f ( x )$$ If $f ( 2 ) = 1$ and $f ^ { \prime } ( 2 ) = 3$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

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

For the function $f(x) = \left(x^{2} + 1\right)\left(3x^{2} - x\right)$, what is the value of $f'(1)$? [3 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

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

Consider the function $f : ( - \infty , \infty ) \rightarrow ( - \infty , \infty )$ defined by
$$f ( x ) = \frac { x ^ { 2 } - a x + 1 } { x ^ { 2 } + a x + 1 } , 0 < a < 2 .$$
Which of the following is true?
(A) $( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) + ( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(B) $( 2 - a ) ^ { 2 } f ^ { \prime \prime } ( 1 ) - ( 2 + a ) ^ { 2 } f ^ { \prime \prime } ( - 1 ) = 0$
(C) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = ( 2 - a ) ^ { 2 }$
(D) $f ^ { \prime } ( 1 ) f ^ { \prime } ( - 1 ) = - ( 2 + a ) ^ { 2 }$

If $f(x) = \begin{vmatrix} x^3 & 2x^2+1 & 1+3x \\ 3x^2+2 & 2x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{vmatrix}$ for all $x \in \mathbb{R}$, then $2f(0) + f'(0)$ is equal to
(1) 48
(2) 24
(3) 42
(4) 18

2. The gradient of the curve $y = \frac { ( 3 x - 2 ) ^ { 2 } } { x \sqrt { x } }$ at the point where $x = 2$ is
A $\frac { 3 } { 2 } \sqrt { 2 }$
B $3 \sqrt { 2 }$
C $4 \sqrt { 2 }$
D $\frac { 9 } { 2 } \sqrt { 2 }$
E $6 \sqrt { 2 }$

Given that $y = \frac { ( 1 - 3 x ) ^ { 2 } } { 2 x ^ { \frac { 3 } { 2 } } }$, which one of the following is a correct expression for $\frac { d y } { d x }$ ?
A $\frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } + \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
B $\frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } + \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
C $\frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
D $- \frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } + \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } + \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
E $\quad - \frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } + \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$
F $- \frac { 9 } { 4 } x ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 2 } x ^ { - \frac { 3 } { 2 } } - \frac { 3 } { 4 } x ^ { - \frac { 5 } { 2 } }$

The function $f$ is given by
$$f ( x ) = \left( \frac { 2 } { x } - \frac { 1 } { 2 x ^ { 2 } } \right) ^ { 2 } \quad ( x \neq 0 )$$
What is the value of $f ^ { \prime \prime } ( 1 )$ ?
A - 3
B - 1
C 5
D 17
E 29
F 80