EXAMPLE Find all function \( g \) such that \( g^{\prime}(x)=4 \sin x+\frac{2 x^{5}-\sqrt{x}}{x} \)

Let \( y=\ln \cos \mathrm{e}^{x} \), then \( y^{\prime}(0)= \)

4. (15б) Выясните, является ли функция \( z=\ln \left(x^{2}+y^{2}+2 y+1\right) \) решением уравнения \( \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 \)

If \( x^{3}+y^{3}=6 x y \), then find \( \frac{d y}{d x} \) and \( \frac{d^{2} y}{d x^{2}} \)

\( \frac{d}{dx}\frac{1}{sin^{2}x} \)

3. (a) Montrer que pour tout réel \( t \geq 0: t-\frac{t^{3}}{3} \leq \arctan (t) \leq t \) (b) Calculer la limite : \( \lim _{x \rightarrow+\infty} x\left(1-x \arctan \left(\frac{1}{x}\right)\right) \).