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. $f'(x) = x$.

The function $f(x) = e^{2x}$ has derivative:
(A) $e^{2x}$
(B) $2e^{x}$
(C) $2e^{2x}$
(D) $4e^{2x}$
(E) $e^{x}$

For a cubic function $f ( x )$ with leading coefficient $6 \pi$, the function $g ( x ) = \frac { 1 } { 2 + \sin ( f ( x ) ) }$ has a local maximum or minimum at $x = \alpha$, and when all $\alpha \geq 0$ are listed in increasing order as $\alpha _ { 1 }$, $\alpha _ { 2 } , \alpha _ { 3 } , \alpha _ { 4 } , \alpha _ { 5 } , \cdots$, the function $g ( x )$ satisfies the following conditions. (가) $\alpha _ { 1 } = 0$ and $g \left( \alpha _ { 1 } \right) = \frac { 2 } { 5 }$. (나) $\frac { 1 } { g \left( \alpha _ { 5 } \right) } = \frac { 1 } { g \left( \alpha _ { 2 } \right) } + \frac { 1 } { 2 }$ When $g ^ { \prime } \left( - \frac { 1 } { 2 } \right) = a \pi$, find the value of $a ^ { 2 }$. (Here, $0 < f ( 0 ) < \frac { \pi } { 2 }$.) [4 points]

Throughout this part, $\lambda$ is a real number belonging to the interval $]0,1[$ and $f, g, h$ are functions in $C^{0}(\mathbb{R}, \mathbb{R}_{+})$ that are integrable and satisfy the following inequality $$\forall x \in \mathbb{R}, \forall y \in \mathbb{R}, \quad h(\lambda x + (1-\lambda) y) \geq f(x)^{\lambda} g(y)^{1-\lambda}.$$ In questions 3), 4) and 5) we additionally assume that $f$ and $g$ are strictly positive, that is, for all real $x$, $f(x) > 0$ and $g(x) > 0$.
Show that the applications $u$ and $v$ are of class $C^{1}$ on the interval $]0,1[$ and, for each $t \in ]0,1[$, calculate the derivatives $u'(t)$ and $v'(t)$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define two functions on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$ by $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta ) \quad \text { and } \quad \widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$$
Express $\frac { \partial \widetilde { f } } { \partial r } ( r , \theta )$ and $\frac { \partial \widetilde { f } } { \partial \theta } ( r , \theta )$ in terms of $r , \theta , \frac { \partial f } { \partial x } ( r \cos \theta , r \sin \theta )$ and $\frac { \partial f } { \partial y } ( r \cos \theta , r \sin \theta )$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define two functions on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$ by $$\forall ( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R } , \quad \widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta ) \quad \text { and } \quad \widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$$
For all $( r , \theta ) \in \mathbb { R } _ { + } ^ { * } \times \mathbb { R }$, show that $\frac { \partial \widetilde { f } } { \partial r } ( r , \theta ) = \frac { 1 } { r } \times \frac { \partial \widetilde { g } } { \partial \theta } ( r , \theta )$ and $\frac { \partial \widetilde { g } } { \partial r } ( r , \theta ) = - \frac { 1 } { r } \times \frac { \partial \widetilde { f } } { \partial \theta } ( r , \theta )$.

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ Prove that, for every positive real $x$, the function $\phi_x$ is of class $\mathcal{C}^2$ on $\mathbb{R}$ and that $$\forall t \in \mathbb{R}, \quad 0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}.$$

We consider two strictly positive real numbers $a$ and $b$, and we set $\rho = \frac{b-a}{b+a}$. We call $\Psi$ the application from $\mathbf{R}$ to $\mathbf{R}$ defined by: $$\forall x \in \mathbf{R}, \Psi(x) = \ln(a^2 \cos^2 x + b^2 \sin^2 x)$$
Show that $\Psi$ is of class $\mathcal{C}^1$ on $\mathbf{R}$, then that for all $x \in \mathbf{R}$, $$\Psi'(x) = 4\sum_{k=1}^{+\infty} \rho^k \sin(2kx)$$

Exercise II

II-A- The function $f$ defined on $\mathbb { R } ^ { * }$ by $f ( x ) = e ^ { \frac { 1 } { x } }$ has derivative $f ^ { \prime } ( x ) = e ^ { \frac { 1 } { x } }$. II-B- The function $F$ defined on $[ 0 ; + \infty [$ by $F ( x ) = x \sqrt { x }$ is an antiderivative of the function $f$ defined by $f ( x ) = \frac { 3 } { 2 } \sqrt { x }$. II-C- The function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = ( \ln ( 3 x ) ) ^ { 2 }$ has derivative $f ^ { \prime } ( x ) = \frac { 2 } { 3 x } \ln ( 3 x )$. II-D- $\quad \lim _ { x \rightarrow 0 } ( x \ln ( x ) - x ) = - \infty$. II-E- $\quad \lim _ { x \rightarrow + \infty } \left( x e ^ { x } - \ln ( x ) \right) = 0$.
For each statement, indicate whether it is TRUE or FALSE.

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
Deduce an expression for the function $m$ and conclude that $m^+ = 0$.
Recall that $m = \psi'$ when $\psi$ is differentiable on $\mathbb{R}_+^*$.

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be functions satisfying $$f(x+y) = f(x) + f(y) + f(x)f(y) \quad \text{and} \quad f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \to 0} g(x) = 1$, then which of the following statements is(are) TRUE?
(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$
(C) The derivative $f'(1)$ is equal to 1
(D) The derivative $f'(0)$ is equal to 1

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g'(0) = \cos(\log 2)$
(3) $g'(0) = -\cos(\log 2)$
(4) $g$ is differentiable at $x = 0$ and $g'(0) = -\sin(\log 2)$

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then: (1) $g'(0) = \cos(\log 2)$ (2) $g'(0) = -\cos(\log 2)$ (3) $g$ is differentiable at $x=0$ and $g'(0) = -\sin(\log 2)$ (4) $g$ is not differentiable at $x=0$

If for $x \in \left(0, \dfrac{1}{4}\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt{x}}{1 - 9x^3}\right)$ is $\sqrt{x} \cdot g(x)$, then $g(x)$ equals:
(1) $\dfrac{9}{1 + 9x^3}$
(2) $\dfrac{3x\sqrt{x}}{1 - 9x^3}$
(3) $\dfrac{3x}{1 - 9x^3}$
(4) $\dfrac{3}{1 + 9x^3}$

If $2y = \cot^{-1}\left(\frac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)$, $\forall x \in \left(0, \frac{\pi}{2}\right)$, then $\frac{dy}{dx}$ is equal to
(1) $\frac{\pi}{6} - x$
(2) $2x - \frac{\pi}{3}$
(3) $x - \frac{\pi}{6}$
(4) None of these

Find the derivative $\frac{\mathrm{d}y(x)}{\mathrm{d}x}$ of the following real function $y(x)$ defined for $0 < x < 1$: $$y(x) = (\arccos x)^{\log x}$$ where $0 < \arccos x < \pi/2$.