Let $f ( x ) = ( 2 x + 1 ) ^ { 3 }$ and let $g$ be the inverse function of $f$. Given that $f ( 0 ) = 1$, what is the value of $g ^ { \prime } ( 1 )$ ?
(A) $- \frac { 2 } { 27 }$
(B) $\frac { 1 } { 54 }$
(C) $\frac { 1 } { 27 }$
(D) $\frac { 1 } { 6 }$
(E) 6

[14 points] Let $\mathbb { R } _ { + }$ denote the set of positive real numbers. A one-to-one and onto function $f : \mathbb { R } _ { + } \rightarrow \mathbb { R } _ { + }$ is called golden if $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R } _ { + }$.
(i) Find all golden functions (if any) of the form $f ( x ) = a x ^ { b }$. Find all golden functions (if any) of the form $f ( x ) = a b ^ { x }$. In both cases $a$ and $b$ are suitable real numbers.
(ii) Show that there is no one-to-one and onto function $f : \mathbb { R } \rightarrow \mathbb { R }$ such that $f ^ { \prime } ( x ) = f ^ { - 1 } ( x )$ for every $x \in \mathbb { R }$.

For the function $f ( x ) = x ^ { 3 } + x + 1$, let $g ( x )$ be its inverse function. What is the value of $g ^ { \prime } ( 1 )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 4 } { 5 }$
(5) 1

There are two functions $f ( x ) , g ( x )$ differentiable on the set of all real numbers. $f ( x )$ is the inverse function of $g ( x )$ and $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$. If the function $h ( x ) = x g ( x )$, what is the value of $h ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) 2
(5) $\frac { 7 } { 3 }$

When the inverse function of $f ( x ) = \frac { 1 } { 1 + e ^ { - x } }$ is $g ( x )$, what is the value of $g ^ { \prime } ( f ( - 1 ) )$? [3 points]
(1) $\frac { 1 } { ( 1 + e ) ^ { 2 } }$
(2) $\frac { e } { 1 + e }$
(3) $\left( \frac { 1 + e } { e } \right) ^ { 2 }$
(4) $\frac { e ^ { 2 } } { 1 + e }$
(5) $\frac { ( 1 + e ) ^ { 2 } } { e }$

For the function $f ( x ) = \left( x ^ { 2 } + 2 \right) e ^ { - x }$, the function $g ( x )$ is differentiable and satisfies
$$g \left( \frac { x + 8 } { 10 } \right) = f ^ { - 1 } ( x ) , \quad g ( 1 ) = 0$$
Find the value of $\left| g ^ { \prime } ( 1 ) \right|$. [4 points]

For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be $$g(x) = f\left(e^{x}\right) + e^{x}$$ The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points]
(1) $\frac{1}{36}$
(2) $\frac{1}{18}$
(3) $\frac{1}{12}$
(4) $\frac{1}{9}$
(5) $\frac{5}{36}$

Let $I$ be an open interval of $\mathbb { R }$. We are given a function $f : I \rightarrow \mathbb { R }$ of class $\mathcal { C } ^ { 3 }$, such that $f ^ { \prime } ( x ) > 0$ for every $x \in I$. Show that $f$ is bijective from $I$ onto the open interval $f ( I )$. We denote by $g : f ( I ) \rightarrow I$ its inverse function. Recall the value of $g ^ { \prime } ( f ( x ) )$. Express $g ^ { \prime \prime } ( f ( x ) )$ as a function of the successive derivatives of $f$ at $x$.

If the function $f(x)=x^{3}+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is

Let $f$ be a real-valued function defined on the interval $( - 1,1 )$ such that $e ^ { - x } f ( x ) = 2 + \int _ { 0 } ^ { x } \sqrt { t ^ { 4 } + 1 } d t$, for all $x \in ( - 1,1 )$, and let $f ^ { - 1 }$ be the inverse function of $f$. Then $\left( f ^ { - 1 } \right) ^ { \prime } ( 2 )$ is equal to
A) 1
B) $\frac { 1 } { 3 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 1 } { e }$

Let $f:\mathbb{R} \rightarrow \mathbb{R}, g:\mathbb{R} \rightarrow \mathbb{R}$ and $h:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable functions such that $f(x) = x^3 + 3x + 2, g(f(x)) = x$ and $h(g(g(x))) = x$ for all $x \in \mathbb{R}$. Then
(A) $g'(2) = \frac{1}{15}$
(B) $h'(1) = 666$
(C) $h(0) = 16$
(D) $h(g(3)) = 36$

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow ( 0,4 )$ be functions defined by
$$f ( x ) = \log _ { e } \left( x ^ { 2 } + 2 x + 4 \right) , \text { and } g ( x ) = \frac { 4 } { 1 + e ^ { - 2 x } }$$
Define the composite function $f \circ g ^ { - 1 }$ by $\left( f \circ g ^ { - 1 } \right) ( x ) = f \left( g ^ { - 1 } ( x ) \right)$, where $g ^ { - 1 }$ is the inverse of the function $g$.
Then the value of the derivative of the composite function $f \circ g ^ { - 1 }$ at $x = 2$ is $\_\_\_\_$.

If $g$ is the inverse of a function $f$ and $f ^ { \prime } ( x ) = \frac { 1 } { 1 + x ^ { 5 } }$, then $g ^ { \prime } ( x )$ is equal to
(1) $\frac { 1 } { 1 + \{ g ( x ) \} ^ { 5 } }$
(2) $1 + \{ g ( x ) \} ^ { 5 }$
(3) $1 + x ^ { 5 }$
(4) $5 x ^ { 4 }$

Let $f$ and $g$ be differentiable functions on $R$ such that $f \circ g$ is the identity function. If for some $a , b \in R , g ^ { \prime } ( a ) = 5$ and $g ( a ) = b$, then $f ^ { \prime } ( b )$ is equal to:
(1) $\frac { 1 } { 5 }$
(2) 1
(3) 5
(4) $\frac { 2 } { 5 }$

Let $f : R \rightarrow R$ be defined as $f(x) = x ^ { 3 } + x - 5$. If $g(x)$ is a function such that $f( g(x) ) = x , \forall x \in R$, then $g ^ { \prime } (63)$ is equal to
(1) 49
(2) $\frac { 1 } { 49 }$
(3) $\frac { 43 } { 49 }$
(4) $\frac { 3 } { 49 }$

Q71. Let $f ( x ) = x ^ { 5 } + 2 x ^ { 3 } + 3 x + 1 , x \in \mathbf { R }$, and $g ( x )$ be a function such that $g ( f ( x ) ) = x$ for all $x \in \mathbf { R }$. Then $\frac { g ( 7 ) } { g ^ { \prime } ( 7 ) }$ is equal to :
(1) 14
(2) 42
(3) 7
(4) 1

For functions f and g defined on the set of real numbers
$$\begin{aligned} & f ( g ( x ) ) = x ^ { 2 } + 4 x - 1 \\ & g ( x ) = x + a \\ & f ^ { \prime } ( 0 ) = 1 \end{aligned}$$
Given this, what is a?
A) $-2$
B) $\frac { -1 } { 4 }$
C) 1
D) $\frac { 3 } { 2 }$
E) 3

For a function f defined on the set of positive real numbers with $f ( 3 ) = 2$, the derivative of the function f is given as
$$f ^ { \prime } ( x ) = x ^ { 2 } + x$$
For the function $\mathbf { g } ( \mathbf { x } ) = \mathbf { f } ^ { - \mathbf { 1 } } ( \mathbf { 2 x } )$, what is the value of $\mathbf { g } ^ { \prime } ( \mathbf { 1 } )$?
A) $\frac { 1 } { 2 }$
B) $\frac { 1 } { 3 }$
C) $\frac { 2 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 1 } { 6 }$