Solt la fonction \( f \) définie ci-dessous: \[ f: x \mapsto(x+1) e^{-8 x+8} \] Déterminer la dérivée de \( f \).

Derivada de \( 2 f(x)=\sqrt{x^{2}+4 x-5} \)

The area bounded by the curve \( y=f(x) \) where \( y=n-\frac{n^{2}}{4}, x=\frac{n}{2} \) and \( X \)-axis equals....... square unit. \( \begin{array}{llll}\text { (a) } \frac{7}{3} & \text { (b) } \frac{3}{4} & \text { (c) } \frac{4}{3} & \text { (d) } \frac{3}{7}\end{array} \)

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

Use the quotient rule to find the derivative of the given function. \( f(t)=\frac{7 t^{2}-9 t}{9 t+5} \)

Find the derivative of the function \( f(x)=-\frac{x^{3}}{x-1} \)