If $y = \left( x ^ { 3 } - \cos x \right) ^ { 5 }$, then $y ^ { \prime } =$
(A) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 }$
(B) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 }$
(C) $5 \left( 3 x ^ { 2 } + \sin x \right)$
(D) $5 \left( 3 x ^ { 2 } + \sin x \right) ^ { 4 } \cdot ( 6 x + \cos x )$
(E) $5 \left( x ^ { 3 } - \cos x \right) ^ { 4 } \cdot \left( 3 x ^ { 2 } + \sin x \right)$

If $f ( x ) = \sqrt { x ^ { 2 } - 4 }$ and $g ( x ) = 3 x - 2$, then the derivative of $f ( g ( x ) )$ at $x = 3$ is
(A) $\frac { 7 } { \sqrt { 5 } }$
(B) $\frac { 14 } { \sqrt { 5 } }$
(C) $\frac { 18 } { \sqrt { 5 } }$
(D) $\frac { 15 } { \sqrt { 21 } }$
(E) $\frac { 30 } { \sqrt { 21 } }$

If $y = \sin ^ { 3 } x$, then $\frac { d y } { d x } =$
(A) $\cos ^ { 3 } x$
(B) $3 \cos ^ { 2 } x$
(C) $3 \sin ^ { 2 } x$
(D) $- 3 \sin ^ { 2 } x \cos x$
(E) $3 \sin ^ { 2 } x \cos x$

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$.

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$ .

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Justify that $\varphi$ is of class $C^1$ on $\mathbb{R}$ and, for every real $t$, give $\varphi'(t)$.

If $f^{\prime}(x) = \sin(\log x)$ and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to (the question continues with answer options as given in the paper).

If $y = \sec\left(\tan^{-1}x\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to
(1) 1
(2) $\sqrt{2}$
(3) $\frac{1}{\sqrt{2}}$
(4) $\frac{1}{2}$

Let $f$ be a differentiable function such that $f ( 1 ) = 2$ and $f ^ { \prime } ( x ) = f ( x )$ for all $x \in R$. If $h ( x ) = f ( f ( x ) )$, then $h ^ { \prime } ( 1 )$ is equal to :
(1) $4 e ^ { 2 }$
(2) $2 e$
(3) $4 e$
(4) $2 e ^ { 2 }$

If $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = 3$, then the derivative of $f ( f ( f ( x ) ) ) + ( f ( x ) ) ^ { 2 }$ at $x = 1$ is:
(1) 9
(2) 12
(3) 15
(4) 33

If $t = \sqrt { x } + 4$, then $\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) _ { t = 4 }$ is:
(1) 4
(2) Zero
(3) 8
(4) 16

Let $y = f(x) = \sin^3\left(\frac{\pi}{3}\cos\left(\frac{\pi}{3\sqrt{2}}\left(-4x^3 + 5x^2 + 1\right)^{3/2}\right)\right)$. Then, at $x = 1$,
(1) $2y' + \sqrt{3}\pi^2 y = 0$
(2) $2y' + 3\pi^2 y = 0$
(3) $\sqrt{2}y' - 3\pi^2 y = 0$
(4) $y' + 3\pi^2 y = 0$

Suppose $f ( x ) = \frac { \left( 2 ^ { x } + 2 ^ { - x } \right) \tan x \sqrt { \tan ^ { - 1 } \left( x ^ { 2 } - x + 1 \right) } } { \left( 7 x ^ { 2 } + 3 x + 1 \right) ^ { 3 } }$. Then the value of $f ^ { \prime } ( 0 )$ is equal to
(1) $\pi$
(2) 0
(3) $\sqrt { \pi }$
(4) $\frac { \pi } { 2 }$

Suppose for a differentiable function $h , h ( 0 ) = 0 , h ( 1 ) = 1$ and $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 1 ) = 2$. If $\mathrm { g } ( x ) = h \left( \mathrm { e } ^ { x } \right) \mathrm { e } ^ { h ( x ) }$, then $g ^ { \prime } ( 0 )$ is equal to:
(1) 5
(2) 4
(3) 8
(4) 3

$$f ( x ) = \sin ^ { 2 } \left( 3 x ^ { 2 } + 2 x + 1 \right)$$
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 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

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