Solve the second order ordinary differential equation \[ \begin{array}{l}\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+l(l+1) y=0 \\ \text { and with the knowledge that Legendre's polynomials are defined such that } \mathrm{P}_{1}(\mathrm{x}=1)=1 \text {, show that } \\ \qquad P_{2}(x)=\frac{1}{2}\left(3 x^{2}-1\right)\end{array} \]

Найти производную функции \( (1-12) \). \( \begin{array}{lll}\text { 1. } 1 & x^{8} . & \text { 2. } 2 x^{-11} . \\ \text { 3. } 2 x^{\frac{2}{3}} . \\ \text { 4. } 2 x^{-\frac{4}{5}} . & \text { 5. } 3 \frac{1}{x^{10}} . & \text { 6. } 3 \sqrt[6]{x^{5}} . \\ \text { 7. } 4 \frac{1}{\sqrt[8]{x^{3}}} . & \text { 8. } 3(1-3 x)^{4} . & \text { 9. } 3(-5 x)^{3} . \\ \text { 10. } 3(4 x-3)^{-6} . & \text { 11. } 4 \sqrt[8]{-5+2 x} . & \text { 12. } 5 \frac{1}{\sqrt[4]{\left(\frac{x}{2}-3\right.}}\end{array} \)

On prend pour première approximation de \( \alpha \) l'abscisse \( x_{1} \) du point d'in- tersection de la tangente au point \( (b, f(b)) \) avec l'axe \( (O x) \), puis on itère la procédure sur \( \left[a, x_{1}\right] \) pour obtenir \( \mathrm{x}_{2} \ldots \) 2. Montrer que \( x_{1}=b-\frac{f(b)}{f^{\prime}(b)} \). 3. Montrer que \( \frac{\mathrm{f}(\mathrm{b})-\mathrm{f}\left(\mathrm{x}_{1}\right)}{\mathrm{b}-\mathrm{x}_{1}} \leq \mathrm{f}^{\prime}(\mathrm{b}) \). 4. En déduire que \( f\left(x_{1}\right) \geq 0 \), puis \( x_{1} \geq \alpha \). \( \quad \) Ceci nous ramène à considérer la suite définie par \( \mathrm{x}_{0}=\mathrm{b} \) et la relation: \( \forall \mathrm{n} \in \mathbb{N} \mathrm{x}_{\mathrm{n}+1}=\mathrm{x}_{\mathrm{n}}-\frac{\mathrm{f}\left(\mathrm{s}_{\mathrm{n}}\right)}{\mathrm{f}^{\prime}\left(\mathrm{x}_{\mathrm{n}}\right)} \) En introduisant la fonction \( g: x \rightarrow x-\frac{f(x)}{f \prime(x)} \) on peut écrire \( x_{n+1}= \) \( g\left(x_{n}\right) \). 5. Etudier la monotonie de g sur \( [\mathrm{a}, \mathrm{b}] \) et montrer que \( \mathrm{g}([\alpha, \mathrm{b}]) \subset[\alpha, \mathrm{b}] \).

Find an equation of the tangent line to the graph of the function at the point \( (1,1) \). \[ y=x^{\cosh (x)} \] \( y=\square \)

Consider the following. \[ g(x)=4 x \operatorname{sech}(x) \] Find \( g^{\prime}(x) \). \( g^{\prime}(x)=\square \) Find any critical numbers of \( g \). (Enter your answers as a comma-separated list. Round your answers to four decimal places.) Find the relative extrema of the function. (Round your answers to four decimal places.) relative maxima \( \quad(x, y)=(\square) \) relative minima \[ (x, y)=(\square) \]

Find the derivative of the function. \[ y=\tanh \left(6 x^{2}-7\right) \] \( y^{\prime}=\square \)